Thank you, Asquared, for your correct answer. We wore hats, not veils. If you forgot your hat, you used some kind of impromptu head covering: Kleenex, handkerchief, glove. It was a mark of respect for the liturgy and for the holy purpose of the church building. Conversely, men removed their hats as a mark of their respect, much as they would remove their hat in the presence of a superior, or a lady, as an act of respect.
The lace doilies did come in as a replacement for the hats that no one wore anymore. By the late 1960’s, no head covering were worn to church any more.
In the earlier 60’s, Jackie Kennedy visited the Pope, and wore a long black lace mantilla instead of the usual hat. It became an instant fashion hit! She also wore a long sleeved dress that reached below the knees–how I wish that would come back into style!