Just having one of the other amps plugged in but not turned on or connected in any other way is enough to cause the hum.
Wow! That's different!
When the amp that is not connected to anything except power is turned on, does the hum get better, worse, or stay the same, compared to when it is turned off?
A possible explanation that occurs to me is that, referring to Figure 1 of the Jensen paper that Jim and I both linked to (great minds think alike :-)), when you plug in the unused amp, and it is in the turned off state, you are placing CPS4 (as defined in Figure 1) of the unused amp in parallel with CPS4 of the amp or amps that is/are being used. That would worsen Vy (as shown in Figure 2) for the amp(s) that is/are in use, thereby worsening the ground loop situation between the amp(s) being used and the preamp or whatever is driving it or them.
If the amount of hum decreases when the unused amp is turned on, it would add credibility to that theory, because doing so would place CPS3 of the amps in parallel, as well as CPS4, which would to some degree reduce the effects of the increase in CPS4.
I presume, btw, that the preamp to amp interconnections are unbalanced, as I would expect these kinds of effects to be much less significant with balanced interconnections.