Start with a small pilot hole and use larger bit. Cut dowel pin off with a coping saw flush to surface. Mark the center of the pin. Drill pilot hold down through pin (Image 1) but not into chair frame. Match drill bit to the diameter of the dowel hole. Drill old dowel out. Check that the new dowel pin is nice and snug. Apply small amount of wood glue to nail and run around inside of hole. Apply glue to half of the dowel pin and tap it in gently with mallet. If new pin is too long, set old dowel pin next to it, mark length (Image 2) and cut off excess with coping saw.