That's actually not true. Hayseeds and Christian fundamentalists were well represented in the GOP from its founding until the 1930s. It was then that the limosine liberal East Coast set deemed them unworthy of recognition and cast them loose.
Most of the fundies were Democrats, being concentrated in the Democratic South and with no social issues to bother them (with the exception of 1928). Hoover appealed to their simple prejudices, but the New Deal won them over- "limousine liberals" from the East Coast (who weren't "liberal" at all, BTW) had nothing to do with it.