another reversal process involving ferric chloride, ammonia & sodium metabisulfite. This procedure doesn't include re-exposing the film.
i've seen reversal processes that don't require light exposure. A stannous chloride fogging bath is one method; another is to use either sodium dithionite (commercial laundry product Iron Out) or sepia toner to fog and develop the halide left after first dev and bleach steps. All of these, of course, can be done in the tank without even opening it.
That said, the few times I've done B&W reversal I've just used light exposure, but never took the film off the reel to do it. I just pulled the reel out of the tank and held it near a bowl reflector floodlamp, moving it around so it gets exposed from all sides. The 800 ft-candle-seconds that Kodak recommends is a minimum; more exposure won't have any significant effect on modern films (the "black sun" effect, where excessive exposure caused
loss of density, only occurred on emulsions from before the 1950s, IIRC), so if in doubt, give the reel a few more seconds under the light. If I were doing it again now (and I might be, soon) I'd probably hit the local Goodwill and try to find an 8" or larger stainless bowl to act as a reflector/concentrator.