I have a Fender California Series P Bass Special... its from the late 90s...has a regular Pbass pickup and a Jazz Pickup in the bridge position. Volume control for the p-pup, volume control for the j-pup and a master tone control.
when the tone is all the way up... there is a hum..more so in the jazz pickup obviously because its a brighter pickup but at loud volumes you can hear it in the P-pup too. Also, when you tap the pickups they sound almost microphonic and make a clicking/buzzing sound.
i checked the wiring and everything seems to be wired right... the soldering looks good, i just replaced the input so that is fine... sounds like it could be a ground issue?? idk though.. there is also a crackling when i turn the tone knob which made me think it could need a new tone pot but would that also cause the hum and microphonic effect in the pickups?
I have a JP 90, very similar except only one volume and one tone with a 3 position switch.
Sounds like you have a bad tone pot, when mine went it was very similar, but the P type pup was the one with more hum. Unfortunately, the Cali Special is a pain to shield due to the J pup being in the body and not under the pickguard. But if you go to the original guitarnuts site and follow the "quieting the beast" instructions you'll do well (It was a breeze on the JP 90 because both pups are in the pickguard). With the JP 90 I had the additional headache of having a flimsy pickguard, which actually flexed slightly when plugging/unplugging and messed up some of the solder joints on the pots adding to the problem. The Cali has a solid guard, so I don't think you should have that problem.
If you're going to replace the tone pot, you might as well replace them all and the capacitor as well.