What creates multiple parties
Today, Brazil has 35 registered parties, and there are some groups which are trying to create new ones. No party has, per se, any majority anywhere, so they are always creating, destroying and changing party coalitions. Over the years, parties were born, fused together, divided into smaller parties, extinguished and refunded, etc. So, it is always a changing situation.
The result of having so many parties is, of course, a big mess in the brazilian politics. What caused that was some democratic reforms that the weak and disorganized, although authoritarian and dictatorial, governement were forced to perform in the early 1980's. There was two parties during the dictatorship, a situation party (ARENA) and an opposition party (MDB). The opposition party were starting to get dangerously powerful and annoying in the vision of the situation party, despite the dictatorship persecution. The opposition was fierce and the dictatorship was unpopular and getting weaker and weaker.
So, the government decided to perform a major political reform. The purpose was to divide the opposition into many small competing parties in order to avoid that any of them would get any significant representativity to pose any danger to the situation party hegemony, and then make them fight against each other instead of effectively fight the government. However, the plan backfired terribly, and the result was that it ended creating a lot of small competing parties both in the situation as in the opposition.
Make USSR break the multipartidarism, without them willing that
So, what would be set in your fictional 1950's or 1960's US was something like that, which would lead to severe and irreversible fragmentation of the political parties. That era was marked by the anti-communist paranoia, so a way to achieve that was some sort of big scandal, much worser than Watergate, which presents severe, large and deep communist infiltration and manipulation rooted inside both of the parties. The cause of the scandal could be someone presenting or leaking convicing evidence that one of the parties was in fact controlled by Moscow, and the scandal escalates quickly enough to show that the truth is that both parties are Moscow-controlled puppets. Some top-secret information leaked from CIA or KGB printed in the first page of a NY Times edition would be able to ignite that, with things just getting worse and worse as the situation progresses. However there should still be an important big quirk/gotcha to prevent that somebody just simply say "f--- that" and hit the big red button leading to WW3.
The public opinion would then perceive that both the Democratic and the Republican parties as traitors. Even democratic and republican congressmen and senators would, with many of them leaving en masse their former parties. As a consequence, both parties quickly disintegrates into a lot of smaller parties which distrusts and infight between them. Defecting/defected deputies and senators from both of the former parties would quickly be able to approve any constitutional changes which would be need to allow an easy creation of new parties.
Those newly created parties will tend to attract people which shares very similar toughts while being hostile to people who thinks differently, even in minor points, very unlike than the present bipartidarian system where each party hosts groups with very different political views. This would quickly lead to a situation where parties have unreconciliable political views and each one of them would oppose the others in varying degrees.
As the years passes, eventually unrelated or distant related reasons would make some of those new parties go extinguished, some new ones born, some of them fuse/rejoin and some or them being further divided. And this would persist for many and many years. With some time, a party would only rarely trust other party, and when that would happen, it would be just on some pontual and occasional situations when they agree more by coincidence than anything else or just want to defeat a proposal or idea from some other third party who both consider as a greater foe.
Finally, something must set in the public and congressmen opinion the idea that multiple parties are better than just two parties because puppeting a lot of small independent parties should be much harder than puppeting just two big ones. Some strong evidence should be presented that somehow, Moscow cannot puppet America's parties efficiently anymore in this new situation and that Moscow wanted to preserve the US bipartidarism system. This is important to keep discredited the idea that fragmentation and disorganization was one of Moscow's objectives, otherwise eventually someone/something might refound the bipartidary system or just start a single-party one.
In this setting, anybody could be your activist. Somebody that told in antecipation that Moscow was controlling both parties and tried to warn America's about that months or even a few years before the scandal is revealed is a likely candidate for winning the presidency after the scandal is revealed. Your activist is much more likely to be some extreme or moderate right-wing communistphobic guy, but it is also perfectly possible that somebody else, even a left-wing (although still anti-soviet) guy could be.