On one I did recently I used white velcro tabs on the rim of the hat and hid the other ends under the puppets white hair. The hat does a great bouncing effect in motion without falling off, and the hair can be combed over them when not in use.

Snail. I was actually wondering if buttons and elastic loops might not be an option. Glad you tried it out. Like Animal31's idea of using the Velcro tabs I think it would help give some movement to the hat without it falling off. I do like the Velcro tabs idea also but have you found that the male side of the Velcro gets caught in the hair or fur?

In my case no, the hat was added/removed before takes. But if you were removing it and putting it back on during a performance then it may get caught up a bit..

So I tried the earth magnets and problem solved! The magnets work great! I hot glued them under his hair, so they aren't visible then I hot glued the other magnets to the interior of his hat. The magnets are strong enough to secure it and gives the hat a cool bobbing effect! Thanks SO much for that tip, it was a life saver!