With Jaguars, it's usually one of two things
- the coils are not wax potted. What this means is the hair-thin wire the pickup is wound with that makes it work is not packed with paraffin wax, allowing the coil wire to move around with influence from the environment, turning it into more of a microphone than an electromagnetic pickup. I'm not sure if Fender potted the CP Jaguar pickups, I do know they didn't do that so some of the lower cost guitars though.
- The claws - I got around this by packing the inside of my pickup covers with that foam egg-crate stuff they ship computer expansion cards in the box with, that and using a pair of pliars to carefully squeeze the teeth "inward" to the coil to make it harder for the claws to vibrate.
You could try taking the claws off to see if they are a part of the cause.
