The smallest thickness of card will change things dramatically, its quite surprising. Just sit the card evenly in the pocket on the body side, so it kicks up the end of the neck. Trial and error is how i found the right thickness to begin with, but now i know just by looking at the action.
This video is usefull..
http://www.expertvillage.com/video/107113_how-make-guitar-neck-shim.htm
Im surprised you cant get your action low on the JS? I've never heard of anyone shimming the neck for this reason on this model. You can usually get it very low by lowering the bridge and a truss rod adjustment.
But.. if you fret your low E string on the last fret and first fret, the distance between the string and fretboard around the 8th fret is how high your action is.
I got mine down to 2mm on my Jaguar, but because of the neck profile (on old Fender design necks) you can end up with 'fret-choke' on string bends with the action too low. This is why companies like Ibanez make flat fretboards to aid low action and speed playing.