As to hats and hat etiquette: Rules and customs may come and go, but basic signs of respect such as removing one's hat in church, during a prayer, or at a funeral service, never goes out of style.
I repeat: unless it's inappropriate to remove it in the first place. Certainly, it sounds like in jlp's tradition, removing the hat would be fine, but let's consider that there are other perspectives that should be honored.
A