I've had good improvements by screening using copper tape. As well as the control cavity and pickup cavities, it might be worth screening the PU covers, fiddly but does help. Also ensuring that the Pickup pole pieces are grounded helps. A strip of copper tape across the back of the pickup should do it but I've found that to get the best connection I used some conductive paint to join the pole
pieces to the tape. All a bit fiddly but gets worthwhile results. Having said that, using both pickups usually is hum cancelling, combined with a mute switch between musical numbers works well, but remembering to unmute before the music starts!!