FDR was a Democrat.

The South was a much different place in the 1930's. The obvious differences are obvious - segregation and Jim Crow laws - but the primary differences would be the people's opinion on government intervention in the economy. The South was rather hard hit by the depression, and was arguably still recovering from the Civil War when it began. People needed work. FDR had a plan to get them that. To top it off, ideas like socialism weren't naughty words across the entirety of the Southern political spectrum. They were quite acceptable, as long as those pesky blacks didn't get anything from it.

The only only, and I mean singular, reason the South turned on Democrats was because of the Civil Rights Act. Once the Democrats were removed from their bases of power, the GOP swept in to take their place. That switch altered the political makeup of the GOP significantly, and likewise caused many conservative elements of the Democratic party to flee to the GOP. Without the CRA, it's arguable that the South would be a far more "socialist" place to live. Segregated, sure, but you get my drift.

The modern hatred of anything attached to the word "socialism" has very little to do with long-past history and much more to do with the alliance of the Moral Majority (read: religious fundamentalists) and big business post-Nixon in the GOP. With the GOP's hold on power in the South rather assured, and the Democratic party being thought of as traitors to the entire South, the new GOP was able to spread its own ideology across the whole region and it found wide acceptance at the ballot box.

Run a search for the GOP's "Southern Strategy" for more reading. I'm sure I've missed a number of things here, but the primary point I'm making is that the political makeup of the South is because of events in the 1960's - not events in the 1930's. It was a much different place before the CRA.

The South was deeply, deeply Democratic. Before the political shift that began with the Dixiecrat party and was in part successfully promulgated by Nixon, winning the Democratic primaries was considered "tantamount to election." So the South voting against FDR then would be like the South voting against a Republican now - Even moreso, as the shift to southern Republicans is relatively recent, whereas pre-WWII a Republian hadn't really had a foothold in the south since reconstruction.

The major independent parties did not spring up in any real significance in the South until post-WWII. The ones that got any margin of votes were usually Northern Socialist or Prohibition parties, neither of which were especially interesting to the South. The Dixiecrat party was not formed until 1948 and it was created in large part due to racial tensions in the Democratic party.

The South and the concept of "small government" wasn't really solidified until segregationists and white supremacists realized "negro race objectives"(from Wither Solid South) were an important part in leftist, big-government policies. FDR's expansion of the government was not as offensive as it wasn't a violent slap in the face to the Southern lifestyle as the Civil Rights Movement and it's subsequent legislation was.