Can moving wire location at circuit breaker reduce hum?

Maybe run an extension cord from the room with the good socket as a temporary measure?

Well, you have received lots of advice but I'll throw one more at you of a bit different nature. If you decide to create a dedicated circuit then pulling wire is plain grunt work that requires no special training. If you can access everything easily you can save a lot of money by pulling the wire yourself. Why pay an electrician to do menial labor? Just be kind and leave him plenty of wire to work with on each end. Especially at the panel. Oh, and please let us know what you find. I believe there is an issue on that particular circuit. 

Get several cheater Plugs and make sure the AC ground is not connected. These are male to female AC connectors that you can disable the AC ground. Many times hum is related to to a ground loop between units. Even if everything connected to the same AC outlet you can still have circulating ground currents.

Get several cheater Plugs and make sure the AC ground is not connected. These are male to female AC connectors that you can disable the AC ground. Many times hum is related to to a ground loop between units. Even if everything connected to the same AC outlet you can still have circulating ground currents.

If the OP already knows that the amps are humming without inputs, then it is not a ground loop.
And. The OP knows that a different circuit does not hum.

These clues are very good, and going against code with a potential for danger would lead me to believe that messing with grounds is likely be best to be avoided.

I would start by getting a big heavy extension cord.  Use the cord to try different outlets.  Before you set about to fix the problem be sure you are fixing the right problem.