I had a similar problem when one of my wires came loose.
A rewire with shielding should pretty much rid the hum.
Dont be put off by "rewire" get yourself a diagram and your sorted. Soldering isn't hard.
Assuming you havent soldered before, just read a quick tips & hints thing off the inet, have a couple of test goes and you should be able to do a clean job (just take your time)
Just make sure you follow the diagram
Dont go wiring the volumes to the tones :P