there's a jag wiring diagram on this site
http://www.jag-stang.com/JaguarHtml/Ima ... ematic.gif
im no expert, but it looks like the strangle switch simply routes the signal through a tiny circuit containing one capactior.
My first though would be that one of the wires either side of the capacitor is bare and is touching something metal in the guitar (control plate? metal shield under the pickguard?).
The metal control plates would pick up and massively amplify any hum..so this will create noise till the whole thing is grounded.
(i'm no electronics expert..but that would be my guess)