You can tip your hat, toss your hat in the ring, pass a hat or even wear several of them. Occasionally, someone will have a bee in their bonnet, or maybe a feather in their cap.
In English, people might say in utter surprise, I'll eat my hat (In German, the same idiom involves eating a broom).
Not too long ago, people in Germany, men in particular, wore hats to protect them from the elements when they left the house. Today, hats aren't a must — but there are still plenty of phrases and idioms involving headwear.