-Double eye turnbuckle (about $3 at your local hardware store)
-Drill and bit
Drill some overlapping holes in a couple of pieces of scrap. You'll put the turnbuckle eyes in the holes in the scrap. Round the sharp edges of the scrap. I used a rasp. Wet the brim of the hat, place the scrap pieces in the hat, the turnbuckle eyes into the holes and expand the turnbuckle. Go slowly. You don't want to end up with a hat that's too big. You can also break the thread in the band if you stretch it too much. Wait for the hat to dry and see how it fits.
Cut the scrap pieces to fit the shape of the hat if you're stretching a nice felt hat. Put dowels through the scrap and the eyes of the turnbuckles to attach the scrap to the turnbuckle.