That's a tall order there. As has been said before, a lot of the reasoning behind the USSR's early lead in Spaceflight came from there R-7 ICBM, which due to it's ignite-all-stages-on-ground approach (necessitated by the serious unreliability of Liquid Rocket Engines at the time) and the heavy Soviet Nuclear Warheads (5.5 Tonnes total), required a massive booster to get the warheads from their Launch Sites to the US. This in turn made retrofitting them to serve as satellite launch vehicles a very simple affair, since they could easily accomplish the task. Heavier LEO payloads and their first BEO payloads were made possible by the addition of upper stages and further improvements to the Core Stage and Boosters. This meant that for them, launching everything up to Soyuz was a relatively simple affair for them and allowed them to achieve their long list of Firsts for a somewhat modest budget.
To help close the gap early on, and make it more even, one way would be to up the mass of US Nucleur Warheads to the point where ICBMs are required early on (or a general favouring of ICBM-delivery systems at an early stage) to permit them a powerful rocket that they can quickly adapt into a launch vehicle and secure maybe a few of the early firsts themselves. It should be noted that the Mercury-Redstone used a Redstone IRBM, while the Mercury-Atlas used an Atlas ICBM.