If you need a real one then that may be difficult, the far left in America was certainly growing a bit more in the 1960s that the 1950’s but it took the Vietnam War to radicalise bits of it and it was a slow process. Johnson took over from Kennedy and his escalation of the war is usually seen as the turning point. That was 1965-ish so it might fit your timeline?
The Weathermen came out of the Students for a Democratic Society but that really was a political movement, which is why some people broke away from it to be more radical.
If there were other, dangerous, underground groups before them, then they were so underground we’ve never heard of them?