I haven’t tried this personally, but I think that this might work.
First a few notes:
This method works only on bolt on necks
This method requires some skill with tools. If you don’t have it, pls stay away from this method, or at least practise on scrap.
You need a router table or similar.
If 1/8” is all you need, then it is easy to find a binding that is wide to make up for this. Just remember to use half f the desired with, or you will end up with a much wider neck. You can also use thin strips of the same wood as the fret board is made from. Gives you a much cleaner and nicer look. My preferred way of binding any fret board, BTW. It gives you almost an “invisible” binding. Very sofisticated.
Take the guitar apart. Lay the neck flat on a bench, fret board up. Measure the neck and, if necessary, shim under the neck/heal so that the neck is perfectly level with the bench. Glue the shims to a larger board to make sure nothing slips wile we do the next step. Find a way to clamp the neck to the fixture we have already built (haven’t really figured it out yet, but it shouldn’t be that hard). Now over to the router table. Use a strait bit and an edge guide to rout away a small amount of wood from the side of the neck. Go all the way up to the fret board, but don’t touch it with the bit. Also stay away from the neck pocket part. Rout away about ¼”. Use a square and a plane to get the sides of the fret board square with the rest of the neck. Otherwise you might end up with a tilted binding. Also be sure to stay away from the neck pocket part of the neck wood or your in trouble. Now you have sufficient support to glue ½” wide strips “under” the fret board. Use the same wood as the neck is made of. The next step is to glue a binding on the ledge that we have created. Blend the new neck wood to the old and you are ready for finishing. This method creates a neck pocket were the fret board have an overhang over the body. It works perfect if the fret board id flush with the body. Otherwise you have to add small strips here too.
The idea of a fret board that overhangs the neck wood might be a bit unorthodox, but I make most of my bolt on necks with a square neck pocket and a similar overhang, pretty much like the Gibson way of making set necks.
Good luck
Peter