I dont know much about Jazz pickups, since I have a Precision bass, but you might want to try some of those Fender vintage p/ups. Ive heard very good things, especially from bassplayer magazine. They arent humbuckers though.
Well, Jazz pickups are wired in a certain way so that if you turn them both on, the hum gets cancelled out. If you just use the neck or bridge pup, it will have the 60 cycle hum. Its kinda like a strat when you can put the switch in between 2 pickups to cancel the hum out. Just turn on both of your volume knobs, and the hum should go away. If you still hear hum, its coming from your amp or some kind of other interference, but not your pickups.
i got both volumes up all the time i will check my cables the guitar player told me to change my pick ups i dont really know a lot about equipment always willing to learn just dont want to spend money on something i dont need i think the bass sounds fine with the stock p/ups.
Breathe, man, breathe! Is it *humming* you hear or is it more of a *buzzing* sound? If the noise doesn't go away when having both pickups on at full, then it could very well be a *RF shielding issue* (unless the pickups are defective, which I refuse to believe when they're on a brand new bass). If so, changing your pickups won't help you, but rather shielding the cavity by painting it with copper or silver paint, or better yet dressing it in copper foil. You can do a search under "shielding" in the Setup forum to see what has already been said about this.