Ah, my favorite kind of problem. These grounding deals can be a trip and a half. Let me ask you a few questions that may help get to a diagnosis. What kind of pups did you have in there before? Were they "dead quiet"? Were they humbuckers or single-coils? What did you do with the extra two wires on your new 4-conductor pups? Is your cavity shielded (with copper foil or some such thing)? What kind of amp are you using? How long is your instrument cable? When you're not touching the switch ring, does the hum level change as you walk around the room? Are you testing this at home or somewhere else? Have you tested your AC outlets to make sure they're properly grounded?
About the ground wire to the bridge, that's a good idea on general principles, however it may or may not help with this particular problem. Basically, if your cable/pickup system is already an antenna, then adding more surface area only results in a better antenna. Although, you might get lucky and hit a node in the transmission line, but the chances of that happening are exceedingly small. My guess is, you'll probably have to diagnose this in a stepwise manner. The first step is to isolate the problem. That's a lot harder than it sounds, 'cause pickups never work in isolation, they're always connected to the controls, the instrument cable, and the amp. A bass that hums wildly on one amp may be perfectly quiet on another. Strange things happen in the universe of hum. Anyway, if you tell me enough about your system I may be able to help. Also if you can post a pic of the inside of the cavity that would be very helpful.
