Get a leather punch, a hammer and a small block of wood. My “Maxi” leather punch set came from Michaels and I use the largest tip with it. (You will punch a hole in the stopper and you want it to be as large as possible. The Mini punch set is too small.)
Place the piece of wood on the floor or a very sturdy table; place the stopper on the wood with the top of the stopper down; center the punch in the inner ring of the stopper and whack it a couple of times with the hammer.
While holding the clay firmly in the stopper, mash down the narrow part on top. You can spread it out as wide as you want. Just don’t disturb the core of clay going through the hole. Trim the bottom part of the clay to about 1/4 inch. Make sure the bottom part of the clay is the same diameter as the inner ring of the stopper. Insert it into the bottle to make sure it will fit, but DO NOT put the stopper all the way in. If you do, you will tear the inner clay core when you try to take it out for baking.You can stop here, but you can make one additional enhancement if you wish. Use a needle tool to make a channel around the bottom part of the clay. After the stopper is baked, place a tiny rubber band in the channel so it will act as an O-ring. You will have to wrap the rubber band twice to make it tight. The tiny rubber bands can be found in a dollar store in the hair accessory section. You will have to experiment a bit to find the proper size for the channel.
You can bake before decorating the top or after. The clay you put on top will now be anchored to the stopper by the core of clay going through the center. If you bake before decorating, use liquid clay to make the new clay stick to the button on top.
Let the stopper cool thoroughly before trying to insert it into a bottle. If the clay is warm, you can tear the inner core by trying to pull it out.