Well, first-past-the-post often tends to favour a strong two-party system (as in the United States) or a "two-and-a-half party system" as in Canada or the UK, there two parties heavily dominates the political scene. I believe that the reason why we do not have a notable third party in the US is because of the separation of powers. You aren't voting for which president you want when you vote for your congressman and vice versa, unlike in the UK and Canada where a vote for your MP is a de facto vote for who you want for Prime Minister. This allows the representatives to be somewhat more maverick-y, if that is even a word, and not allow themselves to be dominated by the top of the party. Add to it that the US has a far stronger history of primaries than both of Canada and the UK and there is far less demand for "we want an alternative to the two parties" since it is much easier to change the parties from within in the US and to make sure that the nominee in your district differs from the "official party line." Furthermore, since the vote for the representative is distinct from the vote for the President, in congressional (and senatorial) elections, the idea of voting for an independent is not a large problem, because it won't lead to any situation of a "minority government that may be destroyed in a vote of no-confidence."
Thus, if you really want to have a strong multi-party system in the US, I'd say that you need to change the electoral system, and you need to do this in a time when third parties would still ocassionally rise up and perform more than just respectably (the Progressive parties of Roosevelt and La Follette, the Socialist party under Eugene V. Debs, etc.) so that more people would be open to the idea. I'd suggest the 1920s-1930s. In the 1930s, the Nebraska state legislature enjoyed a radical reform as the upper and lower chambers were merged into a unicameral body, which has remained the only one of its kind in the United States. This demonstrates that it is indeed possible for changes to the structure of state level government, or that it at least was.
So what would I propose you do...? You should make sure that the US introduces an electoral system nicer to third parties. While I fear that the Scandinavian model of party-list proportional representation is unlikely to be adopted in the US, the Single Transferable Vote may actually go through.
So take a big state, influential, with a streak of political maverickism to it, namely California, sometime in the Depression and have them introduce a bill which makes the California State Assembly elected by the Single Transferable Vote. The effect will be that the Progressive Party will enjoy greater influence in the state government, and perhaps the Socialists will also pick up a seat or two at times. Over time, you could have this spread over the United States, with more and more states adopting it, before finally some Amendment to the Constitution makes the electoral system to the House of Representatives STV.
In the Presidential election, you could then have two candidates nominated by the two "coalitions": the centre-left coalition of the Greens, the Progressives and the Democrats, and the centre-right coalition of the Libertarians and the Republicans.
I freely admit that it would be a stretch of the mind, but it is the best I can offer, I am afraid...