In an area like that your best bet is to clean it with alcohol or acetone. And I mean CLEAN it! I would clean it at least three times using a fresh paper towel each time. Then scratch the surface a little with steel wool or very fine sandpaper and just glue the whole thing down with thin or medium CA

Note, if acetone takes the coloring off of the underside of the trim, it's probably not ultracote - switch to alcohol instead