Describe a La Follette 1924/Hoover 1932 voter

[Blow by blow, the passion dies]

I guess no counties voted this way. Maybe a rural Midwesterner who disliked the New Deal?

[Podgy the Bear]

The rural, dry Midwesterner is probably the most likely one.  In his autobiography, Richard Nixon related that his father Frank (a Quaker and a solid Republican who came from Ohio and moved to California) was a big supporter of LaFollette in 1924. 

He supported Hoover on the wet/dry issue in 1928 and 1932, but Nixon always thought that his father backed FDR in 1936 (because he did support New Deal policies at the outset).

[Orser67]

Perhaps a Republican Midwesterner who was open to left-wing ideas and felt the party had gone in too conservative a direction under Harding/Coolidge, but despised the Democratic Party and distrusted FDR.

[VPH]

A decent number of Midwestern progressive Republicans. For instance, Alf Landon and I believe William Allen White both voted this way.

[Sumner 1868]

Besides the Plains, another example would be pro-tariff union workers in the Upper Midwest. A lot of LaFollette counties in Minnesota and Wisconsin were weak for FDR in 1932 but very strong for him in 1936.