Jordan wrote: “There are exceptions synagogue being one (but then I would presumably remove my bonnet and get a skull cap?) . . .”

Well, when the handful of Jews in my regiment went to shul by army transport, they would wear kilt and tammie while at worship.
Years ago I attended a synagogue service and had not thought to take a hat with me. I was handed a black crepe paper party hat to wear!
Regards,
Mike