Can moving wire location at circuit breaker reduce hum?

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.

Cheap power outlets are plentiful out there. From way back when, I change out every outlet in the house with pro grade Leviton outlets. Same with switches. I have even had to change out some breakers because they can become weak with age.

Even if the particular outlet that your amp is plugged into has no trouble, it is in a chain of outlets, one connected to the next.

Having said this, I don’t think that there will be the described trouble from the outlets themselves, but you may discover some ’interesting wiring’ along the way.

In reference to part of your original question, "Would changing breaker positions closer to the main breaker make a difference?" That is a seldom asked question yet an interesting one. Having worked for Ma Bell in the central telephone office, this was a consideration, at least at the ground buss. We were working with negative 48 volt DC, so this might be apples and oranges in a way. The ’switch’ room, which was the brains of call connections, had it’s very own buss and power for that matter. The intention here was all about noise and circuit protection, but I wonder if the same might be said with AC.

Lastly, I would like to mention the idea of a dedicated circuit for your system. I would agree about running the wire yourself, with at least one thing in mind. The path chosen for the run should ideally be it’s own, not to be shared with other runs or existing wiring (to save some drilling).

Please keep us informed as to what you find as an answer.