Interesting thoughts, really. I have to premise I was born and I live in Italy and my dad teached me the same sign of respect: we had to wear the hat just outside, never inside the home and he used to take off his hat when he met a friend along the street.
About the Church (when I was a little one my family went to the Catholic Church every Sunday) I remember nobody wore an hat, but the old women wore an handkerchief on the head as a sign of respect. There was another sign of respect for women inside the Church: to cover the shoulders.
I'm not a faithful practitioners and I use to pray just by myself (it's a long long story...), but when I have to go to the Church I do my best to give the same signs of respect my family teached me.
Yeah, that sounds about right. I try to be the same way. I'm sure some folks would shake their head at me wearing jeans or shorts to church these days. It's interesting how some norms change, but others stay just as steadfast.