I think you have a ground loop. and when you touch the metal case you're coupling in some noise, not so much because of actual coupling but rather because your ground is floating to whatever you're floating to. So, ground loop: not all "grounds" are the same in your circuit
I see two possible problems(and solutions) here:
-You're not soldering anything to the "ground" (metal ring) of the jack. Your standard 6.35mm jack has both a "live" wire and a shielding. The shielding goes to ground (the long part on the actual jack connector). Since you didn't solder anything to the jack, you have a floating ground there. In some cases, if you get lucky, the case itself will connect all ground pads with a really low resistance path, but believe me, I've worked with high gain (guitar amps) circuits before and it's better to have all grounds connected to one point.
-If you really, really want to insulate the jack for whatever reason, then you can get an insulated jack like
http://www.taydaelectronics.com/6-35mm-1-4-stereo-insulated-switched-socket-jack-solder-lugs.html(that one is stereo, but you should be able to find a mono version. Or just use the stereo and forget about one lug). That would insulate the jack against the casing, but I believe you'd still have the ground loop problem unless you put them all in one point.