glessarch, are you taking an RS tour? From the sound of your prior posts, it appears as if you are going on an independent trip unless you're joining a tour some time before/after Rome and Assisi?
That said, it really doesn't affect your packing either way unless you plan on dining at posh establishments. We almost never return to our accommodations between the time we leave in the early morning and the time we return in the evening so our dinner clothes are our sightseeing apparel: clean, tidy jeans, nicer shirts/tops and broken-in walking shoes. We've never felt out of place in the mid-range, casual restaurants that are our usual choices for dinner.
Your question about hats? Doesn't matter; wear whatever is most comfortable and works best with your personal style. A flat-type cap will obviously pack well in your case whereas a straw Panama style (summer weight) would have to be worn onto the plane to avoid crushing. You'll see a fair amount of baseball caps too. Whatever hat you choose, just remember to take it off when visiting the churches, OK? :O)