A benefit would be that under the sort of circumstance in the OP, assuming sites salted the input passwords, the hashes would need reversing in order to acquire the password and so reuse issues could be obviated. But I don't think that's really worth it when password managers are around.

I'm imagining we have a system where a client signs, and timestamps, a hash that's sent meaning old hashes wouldn't be accepted and reducing hash "replay" possibilities ... but now I'm amateurishly trying to design a crypto scheme ... never a good idea.