I am not sure who would win the nomination. He would have the advantages of incumbency but how much Northern support would he get?
I think Paul V McNutt would fire up the campaign he suspended after the fall of France.
Willkie crushes him and you get a Republican House, though probably not a GOP Senate till 1944.
Well, he would surely have to find some Northern running mate, maybe Joe Kennedy?
Frank Murphy, the former Governor of Michigan, Governor-General of the Philippines,Attorney General would be a very good choice for Garner.
http://en.wikipedia.org/wiki/Frank_Murphy