The symptoms point to an ungrounded bridge.
The ground wire is usually just "pinched" between the bridge and the wood of the body. Since the metal in the bridge and the copper wire are dissimilar metals there is a possibility that corrosion could develop through electrolysis at the connection, causing a less than perfect connection.
You can do a quick and dirty test by simply attaching a temporary ground wire between the bridge and the attaching nut on the output jack. If the problem goes away, or improves, remove the bridge and scrape the ground wire clean and shiny, sand the bottom of the bridge plate where it contacts the ground wire to clean the surface and reassemble everything.
If your bass isn't shielded, it may be a good time to do so. It can never hurt to do a good shielding job.
If your bass is not "star" grounded, it's an easy mod that definitely makes for less noisy electronics. Star grounding is basically just bringing all of the grounds to one common point, usually, the ground tab on the output jack or the back of one of the pots. There's lots of info on the web with a more thorough description, if you decide to go that route.
Star grounding helps to eliminate "ground loops", which are very hard to diagnose because the ground connections can be electrically sound but still cause AC hum.
Good luck with the problem.