If you look at Congressional Elections, it was a pretty normal thing from the 1910s until the 1950s for there to a gaggle of minor party members of congress. Progressive Party, Farmer-Labour Party, Nonpartisan League, Socialist Party, American Labor Party, the occasional Independent, Prohibition Party, etc.
What happened? Was it the Democrats sucking the left-energy out of the room? That didn't seem to be an issue during the New Deal heyday.