[Phlogiston]

If it gets louder as you increase the vol on the set (not changing with input, MP3, vol changes) then yeah, it is probably picking up the hum in the audio stages, could be those tone controls and yeah I think that at lest one is shielded - green wire, I think. Many of these issues can be found by pushing a few wires and such around with your hand, but be careful.

You could use the MP3 input (switched?) and even ground (short) the input so that you get no nasty ,real, input to distract from your noise. Then you can also turn the volume up to make hearing changes easier.

I just pulled the tone/on/off button assembly out of my 12-S-568 (1941) and see that I replaced the shield that goes over all but the 2 power/switch wires. The button assembly is grounded to the shield. You could also have one of the old, crusty wires shorting to the shield, or even a new, not so crusty wire doing so. These switch assemblies can be a dirty contact bonanza.

Other than an induced hum, the other possibility is increased loading of the power supply, beyond what would be normal for increased vol (minimal difference). But the most common - bad bias on the output tube, should be constant and unchanging with vol.

Using a signal sniffer might help but they can induce hum as well.
Russ