I have done my first wiring repair. It was on my Pee Wee Les Paul, which, with the help of a nut extender, is one of my lap steels. The white wire came off the back of the volume pot, so the buzz was horrendous. And now it's mostly gone.
The action on the the volume pot is kinda weird, so I think a fuller rewiring is in the cards, but as a warm-up to the job I'm going to start on my Tele, I feel very good. Thanks to Seymour Duncan for the wiring diagram (simple as it is) and to Patrick for the iron and solder.