If the area is easy to sand, I like to use CA and push about half a millimeter extra through the hole. When the CA is set, sand the ends flush to the surface.
If you use that method, you have to mask off the ends of the fiber before painting.
Otherwise, use fine sandpaper and make a nice flat end on the fiber and push it into the hole. Glue it from the back with a dab of CA or 5 minute epoxy