Do I need a new bumper for 37s?

You shouldn't need a stubby bumper for 37s, no.

I don't think you'll encounter any rubbing issues at all with the bumper. I could be wrong, but I don't think so.
 
If one is worried about the rubbing, I would look at the Smittybuilt Carbine model. On mine, it will still give me clearance to mount the Class 3 Curtis hitch I have coming in. Considering that I have 35's, the 37's will need room to make that turn or even departure point in an off-road trail.
 
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