I think you're right about the Christian origin, and it isn't only Catholic. My SO and I were talking about it, and we said we had each seen a change since our childhood in the 1950s, when it was inconceivable that a woman enter church (RC in her upbringing, Protestant in mine) without a head covering - a scarf or a hat. This is not as strict an unspoken rule today.

And the biblical injunction on which this was based?

St Paul, 1 Corinthians, King James version

3: But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
4: Every man praying or prophesying, having his head covered, dishonoureth his head.
5: But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven.
6: For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.
7: For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
8: For the man is not of the woman; but the woman of the man.
9: Neither was the man created for the woman; but the woman for the man.
...
13: Judge in yourselves: is it comely that a woman pray unto God uncovered?
14: Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him?
15: But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.

And what was this a fashionable fossil form of?