[morzh]

Last thing,

Have you tried grounding the chassis? Your filaments are all grounded on on end. Wonder if this would help.
Simple touching the chassis with your finger reducing the hum would tell if it could be a problem.

Any Ground (chassis) paths going through rivets that are loose/rusty?

Also see if by any chance the Ground path for filaments is used (tapped in the middle from transformer to filaments) for grounding other parts, like cathodes etc.