There are more meaningful teachings on the head-covering in the Bible than the cultural background.
Men should not wear any hat or any other type of head-covering like Jewish Kefa while they pray or prophesying ( like sermon).
Women should wear the head covering during the praying or prophesying.
Read the Bible:
1 Corinthians 11:
1 Be ye followers of me, even as I also am of Christ. 2 Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you. 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. 10 For this cause ought the woman to have power on her head because of the angels. 11 Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. 12 For as the woman is of the man, even so is the man also by the woman; but all things of God. 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. 16 But if any man seem to be contentious, we have no such custom, neither the churches of God.
1) Paul explains the reason for the Head-Covering for the women, No-covering for the Men very well, not based on any cultural reasons, but based on the principle of Creation. If men cover their heads, it covers the glory of Jesus Christ. If women covers their heads, it covers the glory of men so that only the glory of Christ be demonstrated.
2) Some argue that the long hair can replace the covering as we read in v 15. Verse 15 is abused by many translators, but the word behind is "ANTI" which can be interpreted as 1) Opposition, 2) Replacement 3) For the sake of, 4) On top of, In addition. In this case, 4) th meaning applies. Otherwise, you will find the apparent contradiction with verse 6.
4th interpretation is found in John 1:16 ( Grace FOR Grace).
So-called Plymouth Brethren practices exactly as this Bible teachings, which is not difficult to obey.
Moreover, if you look at the photos taken before WWI, mostly the women attending the churches or visiting the offices wore the hats. Women's no-headcovering has just started since the industrialization and the employment of the women workers.
Jewish Kefa and Catholic head-covering for the clergies ( or so-called Monsignors) are WRONG!