In "The Quiet Man," Maureen O'Hara just pulls her shawl over her head when she wants to go into the church.
I did the same thing in Rome, with one of those pashmina shawls. With churches around every corner a veil was impractical.
While I was looking for the photos of Irish women wearing shawls that I posted above, I ran across a web page about Irish Fashions that said pashminas were all the rage for a while in Ireland.
A few years ago at Christmas I went on-line and bought a couple dozen pashminas for all the women in the family. Several of them, at least, are still being worn each Sunday.
What I like about pashmina shawls and similar items is that it makes covering your head seem like a natural part of every day life, rather than a circular doily-type chapel veil that you pop on your head as your genuflect in the aisle.