FDR still wins handily, but Garner carries a few states in the South and takes a away a few thousand votes away from the Democratic ticket that allows Willkie to take a few more states with pluralities.
✓ President Franklin D. Roosevelt (D-NY)/Secretary of Agriculture Henry A. Wallace (D-IA): 367 EV. (51.93%)
Businessman Wendell Willkie (R-NY)/Senator Charles L. McNary (R-OR): 126 EV. (44.27%)
Vice President John Nance Garner (I-TX)/Senator Allen J. Ellender (I-LA): 38 EV. (2.72%)
While I agree with you that garner would get a couple of states, I think him being vice president would have had a more valid track record than thurmond being a governor, and would therefore probably give him a good 10 to 15 points more in the southern states he loses, would would net him 3.5 to maybe even 4 percent of the vote