Off axis sound - Letting it bleed

What mics are known for good (as in, not "colored") off axis sound? I ask because I usually record live sessions that can be fairly loud in my room, and I have done about as much as I can in terms of placement of amps, gobos, etc.. I am learning to "let it bleed" since that is what I'm dealing with.

I have read that SDCs are generally better than LDC or dynamic for off axis sound. Does hypercardioid as a pattern have any benefit in this area?

Things like DPA and Schoeps are out of my league. I may check out a few mic options if I can find some in the < $300-500 range. For example, Shure KSM 137 or Beyerdynamic M201 (not SDC but pretty small diaphragm and hyper).

If you record loud live band sessions and are letting it bleed, I'm curious to know what works for you...