Hum can be the result of interference from other electrical objects. Even a battery powered watch can cause minor buzzing. The one sure way to determine what's causing the hum is to have someone go through the guitar and test everything one-by-one - wiring (grounding), cold solder joints, capacitors, and few other things. Will the hum disappear when you touch the strings or bridge? If it goes away, you might think of shielding the cavities and making sure the grounds are properly connected. Bad grounds are the most likely source. If you bought it used, the PO might have tried to make some modifications (and failed). If it was purchased new, the shop where you bought it should check it for no cost. In fact, just a "hum search" at a guitar shop shouldn't cost much, and either should any repairs. A shielding job might take a little more time to complete, but you can do it yourself with shielding paint or copper foil, if you have the skill to remove the "guts".
It's probably a bad capacitor allowing AC to get into the signal path, or one of the power tubes is bad. The fact that it happens when the volume is off means that it's not the preamp section.
With or with out a guitar plugged in
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