Jewish dating customs
Among the Greeks it seems that men did not ordinarily wear anything on their heads for worship of their gods, or in public generally.
But the Greek headcovering customs for women during the first century are hard to determine with any degree of certainty.
We should beware of putting too much weight upon this evidence, however, because it may be that in these illustrations the women are depicted without headcoverings because they are at home, and perhaps it was merely a convention of Greek art to portray women in this way.
It is hard to tell from the depictions alone whether or not the women are in a public setting.
The more research I have done on the subject, the more I have come to see how problematic is the widely-accepted interpretation of 1 Corinthians 11:2-16 which asserts that Paul is merely urging the saints in Corinth to conform to local and secular customs.
In my opinion this idea raises more problems than it solves.