To create a massive new space race I say have the USSR get its act together in the early 60s before Korolev dies and have their main rocket engineer work out a compromise with the main rocket designer. The N-1 has something like 33 engines to the Saturn V's 9. Make a program where the USSR has real space launch capabilities that could toss a manned craft to the moon and there you go. America to the moon in 1969, USSR to the moon in 1972 then to Mars in 1973 to celebrate the 55th anniversary of the Red Revolution on the Red planet. Forget military spending at that point, and forget the welfare system in the US, space becomes the new propaganda battleground and maybe we would have seen 2001 actually look like the movie of the same name. Alternatively you could have the USSR in its death throes stave off collapse temporarily under a very early Gorbachev by launching a Soyuz with a modified L-3 attached via an Energia rocket. This might to kick-start another space race but I'm not sure how long it would last.