However, since some shielding does help it is obviously not induced at the input side.
I find it hard to believe that the unit was sold like that and you are upgrading a faulty original design. I wonder if the big copper strap around the transformer (can be internal) has cut loose. They are soldered and if this thing is new enough to use lead free solder I think that a viable suspect.
Can you see the salesman in the store saying "Yes it has a reverb but you can't use it because of an annoying hum caused by a design defect" ?
Anyway, I would have a good look at the reverb unit itself. If this thing isn't junk, it has to have some sort of shielding from OTHER sources of EMI no ? What, you have to keep the thing three meters away from anything with a power transformer ? I tend to doubt it. I think inspection of the reverb unit might reveal a fault there.
+++++
It is a regular complaint by owners of these amps. May be not so obvious when they are used in 60Hz countries, 50Hz here. The next time I came across one I might explore with small pieces of mu-metal around the coils in the tank, and put a scope on the thing. Reading around mu-metal it seems that to restore its invisibility cloak effect then you just need a hydrogen furnace to re-anneal, I'll knock one up from my pile of scrap