Many denominations accept without question that a woman should wear a hat in Church, or even wear some kind of bonnet at all times. The justification for this is said to be found in Corinthians 11:
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.
Paul here talks about a “covering”. The word “hat” is never used. The Greek used is katakalupto. Kalupto means covered, the prefix kata suggests “well covered” or “going down”. This is often taken to mean a veil, in which case a hat does not fit the bill. But 80% of the way through this passage Paul suddenly starts talking about “hair”, with no indication at all that he is switching to a different topic, or why. The simple answer to this apparent change is that there is no change. Paul has been talking about hair all along. He even says “ her hair is given her for a covering”. He also asks “ Doth not even nature itself teach you” that a woman should have long hair and a man should have short hair? Nature itself does indeed teach that. Even for non-Christians it seems simply natural that women should have long hair and men should wear it short. Even though pop groups popularised long hair for men in the 1960’s, it didn’t catch on. Even though feminism and fashion promotes similarity of hair and dress for men and women, the long/short, female/male division remains the norm. Paul is actually saying something very simple, that the God-given differences between men and women should be preserved, not destroyed. Also, earlier in the passage Paul refers to it being a “shame for a woman to be shorn or shaven”; this too we should recognise as obvious. Following wars, women accused of having relations with the enemy have often had their hair shaved. For a woman it is an obvious sign of shame. In short, long hair is the only covering a woman needs.
The implications of this go well beyond the wearing of a hat in church. If a woman who is Saved wears a hat in church because she believes that is right, it is not necessarily a problem for her personally. However, the fact that this plain and simple passage is misinterpreted so consistently by so many churches shows that they are not following Scripture, but tradition. It is but one example of where the Churches and Authentic Christianity divide.
On the individual level, any person who is genuinely Christian should be leaving behind these misinterpretations as they work out their Salvation; any person whose Christianity is merely cultural will remain content with the traditions, and content with Church.
See Sorting Churchgoing Christians from Churchgoing Churchgoers for more on the divergence of Church teachings from the Bible, and its implications.