Shielding is definitely the first step. While I might use shielding paint (the usual Stewmac stuff, not silver), for the pickup routes for asthetic reasons, I'd use copper tape in the control cavity. I'd also use copper tape on the underside of the control plate.
If, after shielding, there is still too much hum, the next step would be shielding the wire from the pickups to the controls. Also, does the bass behave the same in all different environments or just in your house or other specific places? Some houses seem to just have wiring that disagrees with single coil instruments. Also things like dimmer switches and other rheostats are notorious for interfering with SC PUs.
The next step you might consider is adding an internal dummy PU, (coil with no magnets), not unlike Alembic's hum canceler on Series instruments. I haven't looked for basses, but there are several youtubes on how to do this with a Strat.
Bill, tgo