I've made several of these and haven't experienced that problem. Be sure you use the foam disk that comes with the kit. Put it on the inside of the hat at the top below where the animal head sits on top. Then, using a yarn needle, thread the ribbons attached to the bottom of the head through the hat and through the holes in the foam disk. Tie the ribbons tightly (in a bow so that you can untie them to wash the hat).
I've made the puppy a couple of times and haven't encountered this problem. My advice is the same a fshinbaum.