2. In some communities, a hat and jacket are considered part of the requirement for dressing nicely for davening.
So as not to make the same mistake, what type of hat (and jacket) do you mean?
Or, just use a brown paper bag, or drape a towel over your head.
What is wrong with a Fez or a turban? If you drape a towel over your head, is it then a tallis?