In 1932 - 3 Socialist-Communist candidates took about 3% of the vote. In 1936 - the Union Party took 2% and Socialist-Communist parties took less than an additional 1%.
http://en.wikipedia.org/wiki/Union_Party_%28United_Stat... You can make a really vague argument that Roosevelt was enacting rural reforms that had been part of the Populist movement - but you can also make an argument that the Republicans inherited many of those ideas through the lens of isolationism which stood in stark contrast to Wilsonianism.
Either way - the claim that anti-Democratic-Republican parties had growing strength is statistically false.
Many observers at the time felt that there was a place for a party more radical than Roosevelt and the Democrats but still non-Marxist in the political spectrum of the time.
Rumored political aspirations of Huey Long
Although many people expected Huey Long, the colorful Democratic senator from Louisiana, to run as a third-party candidate with his "Share Our Wealth" program as his platform, his bid was cut short when he was assassinated in September 1935.
It was later revealed by historian and Long biographer T. Harry Williams that the senator had never, in fact, intended to run for the presidency in 1936. Instead, he had been planning with Father Charles Coughlin, a Catholic priest and populist talk radio personality, to run someone else on the soon-to-be-formed Share Our Wealth Party ticket. According to Williams, the idea was that this candidate would split the left-wing vote with President Roosevelt, thereby electing a Republican president and proving the electoral appeal of SOW. Long would then wait four years and run for president as a Democrat in 1940.
Prior to Long's death, leading contenders for the role of the sacrificial 1936 candidate included Senators Burton K. Wheeler (D-Montana) and William E. Borah (R-Idaho), and Governor Floyd B. Olson (FL-Minnesota). After the assassination, however, the two senators lost interest in the idea (Borah ran as a Republican, garnering only a few delegates and losing the nomination to Kansas governor Alf Landon) and Olson was diagnosed with terminal stomach cancer.
http://en.wikipedia.org/wiki/Union_Party_%28United_Stat... A radical is a man with both feet firmly planted — in the air. A conservative is a man with two perfectly good legs who, however, has never learned to walk forward. A reactionary is a somnambulist walking backwards. A liberal is a man who uses his legs and his hands at the behest — at the command — of his head.-Franklin Delano Roosevelt - Radio Address to the New York Herald Tribune Forum (26 October 1939)