Hmmm... 😕 I'm just trying to throw around some suggestions for you, but am running out of ideas fast...
By chance I re-strung my own JS this AM: had to cos the guitar has been lying around for months completely unused with an old set of 12s on, and I'd failed to slacken them off... Took them off completely to lemon oil the fretboard and adjust the bridge height. That's been done with the low E side set slightly higher (just check yours again and make sure the screws are okay). Like a fool, I nearly put the new set of 11's on upside down (that's why I asked if that's what you'd done). The bridge needed slackening off (it gets an occasional wobble) - intonation is as near as dammit. The guitar sounds great plugged in, but as the other guy says - Jag-Stangs do sound rattly when un-amplified.
Just going to wait a while for the neck to settle down with the new strings on, then check the truss rod adjustment using a capo and feeler gauges. Good luck!!
If you've done all of the above and still have problems, then hopefully someone else will chime in with better ideas!