I double checked; this challenge has indeed been posted in the Pre-1900 Forum.
One could hardly tell since just about every reply seems focused on post-1900 changes!
I hardly care if the topic sprawls across the arbitrary boundary, and this particular one would be most likely to. However the amazing thing that is there is no weight given to pre-1900 formation or roots of precursor parties or anything much happening before the election of 1896!
I was going to throw in a mention of the multi-party system that develops in Jonathan Edelstein's
Malê Rising TL, which has recently been finished. It "fails" to meet the challenge--only because there winds up being not just one leftist party but two or more at various times--without any drastic or anyway essential change in the election rules, the US forms a fairly stable four or five party system! At least two of those are different flavors of left-wing, certainly by OTL post-Cold War era standards! (By those standards perhaps all but one of the parties are leftist).
Note that in the 19th Century OTL, a Left-Wing Third Party did form in the USA--the Republican Party. It managed to form and grow despite the existence of a two-party system between the Whigs and the Democrats, and by any sensible definition was at the time of its emergence (and first half-decade or so of dominance, under Lincoln) quite evidently left of either former party. It moved to the center (or rather, pretty much defined the "center" by definition) but various splinter parties formed mostly from dissidents who were former, foundational Republican stalwarts--and these dissidents were invariably both to the left of the emerging plutocratic-oligarchic party bosses, and quite of the mind that the GOP was deserting them, not the other way round.
So, with the example of the formation of the GOP before us and noting the ferment of dissent that roiled the apparent hegemony of that party throughout the period 1870-1890, I'd think there ought to be a lot more discussion of potentials of either forming the leftist one-of-three party in the late 19th century, or anyway considering how alternative courses of the various radical movements of that period might have laid the groundwork for a leftist third party to form any time in the 20th century.
OTL of course the People's Party bid fair to nearly meet the challenge all by itself. Had the Populists been better able and willing to link up with non-agrarian (or mining) working-class movements, to ally with urban industrial labor radicals, I'd think they might not only have survived but tipped the balance to become the dominant party some time before 1920. I'm not talking here about Marxist revolution either; the hegemony of such a party would be checked and opposed by the conservatives and I'm talking about scenarios where the radicals remain within bounds of the American constitutional system enough that they can be voted out of power eventually. But just as the radicals could or anyway would not move to crush the potential of the conservatives to unseat them, so the conservatives could not get away with reversing everything the radicals do. The center of mass of US politics is moved leftward and the new Rad-Pop Party alternates in periods of dominance with one or two other parties.
Jonathan's Malê Rising TL has a lot of fans, myself being among them; perhaps the model he presents for a US multipolar party system is not highly realistic. But it did seem to emerge gradually and naturally (and without any preliminary reform of basic election procedure to favor multiple parties--it happens under the current Electoral College and pervasive winner-take-all local electoral systems). Perhaps a crucial element in the process was a more limited sweep of the post-Civil War Amendments. (Lincoln is not assassinated and serves out both his terms and the Republicans are less Radical-dominated than OTL). In addition to weaker (initially) Federal mandates of broad civil rights, it is also essential that a few regions (mainly South Carolina) are dominated by local African-American majorities and remain bastions of full civil rights for AA's (though the majority of the South and to an extent the North and West are at least as reactionary as OTL, in some places and some ways being worse for some time). So the struggle for basic civil rights becomes a grassroots and local one, with at least one State of the Union being a bastion of the generally losing side. I suspect this fact of the ATL helps lay the groundwork for minority coalitions of all kinds persevering as independent movements.
Another step on the road was the ATL Great War, happening some twenty years before our OTL WWI, and the debate in the USA about whether to join it and if so on what side with what war aims, which creates space (along with a generation's worth of general grassroots organizing around various radical causes) for a Peace Party movement, which was never very large in headcount of reliable voting publics, but did tend to command the allegiance of very illustrious leading citizens (such as say, Samuel Clemens and Harriet Tubman, in alliance with yet other interesting figures). In this context, two larger and more stable left-wing parties also were forming--separate Progressives and Populists. Jonathan rather confused me by somewhat reversing the OTL organizational histories of these two movements--the Progs (and I admit this still seems a bit odd and perhaps improbable to me) form from moderate but definitely reformist elements of both the Republican and Democratic parties, fed up with being filtered out of influence by the mainstream party machinery, and joining together under a new party banner to champion pretty much the OTL Progressive agenda of reform from above by middle and upper class elite elements. (OTL people of this mind never formed such a third party, instead operating within both the Republican and Democratic parties, depending on which dominated their local region). Whereas in the ATL the People's Party, a union of radical agrarians and urban labor advocates from the beginning, begins as movements within the established two parties but only gradually realize they must also split off into their own separate party.
In any case the elections of the early 1890s, polarized and intensified by the war question as well as accumulated tectonic forces built up by the general climate of popular empowerment that is the spirit of the time line (radiating from a radical ATL figure, Paulo Abacar, after whose movement in West Africa the TL is named--radiating through the USA in particular via an extra level of organized rebellion against slavery by African Americans, which helps explain how the South Carolinian black majority winds up prevailing in their state) become fractured, with the election returns giving no clear majority for President while the houses of Congress must seat representatives of all the new "third" parties, so any horsetrading to settle the Presidential race in Congress must still balance the interests of at least two or three of four or five effective factions. Whereas there is enough common ground cross-cutting across the factions, which differ in emphases and policy on particular issues rather than radically across the board, that effective coalitions are possible.
Once the precedents of settling multi-polar election results are set and the basic conditions that enable third and fourth parties to make a respectable showing that translates into meaningful power sharing persist for a while, the multipolar system becomes customary. It is unclear to me whether there was any substantial reform in the basic US election machinery to make it more favorable to multiple parties or not.
As a general rule I'd think that such reforms would follow the rise of a stable three or more party system rather than lay groundwork for it, because under the two-party system the dominant two benefit tremendously from machinery that tends to eradicate the visibility of rival new parties. However it might be, as Jonathan's example would indicate, that a multipolar party system can survive despite rules that would tend to reinforce a two-party system, and if it has done so for a couple generations reform of the basic election rules might not be a very popular cause. I myself would love to see them, but getting that result without first forming a third party that is strong enough to stay in the game a long time but not so strong as to simply destroy one of the formerly dominant two seems Utopian to me.
Without the perhaps miraculous intervention of the thought of Paulo Abacar to redeem our other ATLs, how are we to achieve realistic leftist third parties that come to stay in the USA? Well I just want to close this post by reiterating that there is plenty of material to work with in the OTL ferment of the second half of the 19th century. And that a strongly leftist party will tend to be unable to completely replace either of the two established parties, but on the other hand could have a bastion of stalwart voters who will never cross lines to support either of those. Such a leftist party might fluctuate into the position of being the majority party from time to time, and in recession might maintain via its bastion of reliable supporters plus specific candidate personalities and specific electoral issues, keep open a revolving door of state legislature dominance and/or governor's positions. while maintaining a strong if third-ranked (often I think, perhaps second-rank) position in the House and probably manage to always have at least a handful of Senators as well. As long as the leftist party can maintain some visible level of presence, by retaining control of some states or anyway being the dominant opposition party in some of them, and keeping their hand in Congress on a significant level, as well as a round of control of city and other local governments, its principles--the same ones that generally block it from achieving a solid majority across the board--can guarantee it retains a solid base the other two parties can't erode, and so remains a contender for electoral victory at all levels. If this is the case, then while some or perhaps essentially all non-leftist politicians may make crusading against them the center of their political position, at least some of the time at least some of the politicians of the other two parties will find it tempting or even congenial to form coalitions with the leftists, and both cooperating parties might profit much from this. Thus, even if the leftist party can only rarely or perhaps never elect a President, they still can wind up having a solid amount of influence proportional to how many votes they tend to pick up, even if those votes often fail to elect a candidate.
So, if I were challenged to produce any third party at all, and charged not to make that third party a regional one, I would point to the left-wing constituency as a likely basis for a party that may only rarely emerge from third place but can remain standing in a strong third place position for generations. The conflict of interest between the dominant capitalist interest of the USA as we know it prevents them from taking power comprehensively, but the conflict of interest between dominant capitalism and the working classes can also generate a constituency militant enough to persist in organizing separately despite frequent defeats at the polls, but compliant enough with US society as it is to work within the system on these terms. "They'd rather be right than President!"
Of course, they might also wind up producing a President someday, and perhaps become the dominant party after all, and redefine the terms on which rivals campaign to replace them.
