There were a number of objectively conservative parties that backed Hindenburg: Catholic Centre Party, BVP, DVP, and DStP. Hindenburg chose to support Hitler because of the threat posed by the left to the bourgeois interests he represented and because Hitler didn't really challenge said interests.
The SPD also chose Hindenburg over Thälmann, and if they knew he was going to support Hitler, then maybe they would've acted differently. But either way they weren't Hindenburg's core base of conservative support.