I had a similar problem with my Epi Les Paul when I got it out of the attic a few months back.
With me, it was the bridge pickup failing.
I tried all sorts of things, and then I noticed that when I switched to the pickup with the big loss in volume (not the middle position), if I tapped the other pickup with a screwdriver it was was still active.
So I took the switch out to see if anything was looking dodgy. I found several years' worth of cr@p in it. I gave it all a good going over with Servisol switch cleaner and the problem went away. I think there was some sort of short or lack of connection in the switch contacts themselves - I didn't fiddle with the wiring or soldering. It was still a little flakey sometimes after this - but my jack socket and plate were on their last legs anyway, causing even bigger problems!
After I put Mules in a month or so back, I decided to replace all the electrics last weekend - no problems at all now. (And by the way, the Mules sound SUPERB now, an unbelievable difference with decent pots and caps).