Use a proper stripping tool. Don't nick the wire, especially for single core stuff.
Hold the wire along the route you need it to go, cut to length, strip the ends, tack one end, tack the other, finagle into place with pliers, tweezers, screwdriver, fingernails, whatever. Teflon is no good for gluing but the other kinds are. A blob of glue (anything that's good for filling gaps, and has the consistency of, say, nail polish, will do) helps hold things in place, especially important for commercial bodges.
The pro glue I think is UV cure resin. Dab it on, shine a UV LED on it, and it's solid. Often in green, used for patching soldermask.
Tim