These messages arguably do not contradict. They relate to modesty and women should cover themselves and dress modestly. The veil / covering of hair was usual practice at the time and Nuns still cover their hair / head.
1 Corinthians 11:5-15 – itself appears contradictory. V5 relates to women praying / prophesising – yet v6 appears to make a general point of covering. V15 could be seen as contradicting v6 – but is more likely related to general point that women should keep long hair and men short hair see v14. Also Ezekiel 44:20
1 C 11:5-15 - 5 And every woman who prays or prophesies with her head uncovered dishonors her head--it is just as though her head were shaved. 6 If a woman does not cover her head, she should have her hair cut off; and if it is a disgrace for a woman to have her hair cut or shaved off, she should cover her head. 7 A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. 8 For man did not come from woman, but woman from man; 9 neither was man created for woman, but woman for man. 10 For this reason, and because of the angels, the woman ought to have a sign of authority on her head. 11 In the Lord, however, woman is not independent of man, nor is man independent of woman. 12
For as woman came from man, so also man is born of woman. But everything comes from God. 13 Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered? 14 Does not the very nature of things teach you that if a man has long hair, it is a disgrace to him, 15 but that if a woman has long hair, it is her glory? For long hair is given to her as a covering.
Man should trim
Ezekiel 44:20 ESV / 123 - They shall not shave their heads or let their locks grow long; they shall surely trim the hair of their heads.
1 Timothy 2:9-13 9 I also want the women to dress modestly, with decency and propriety, adorning themselves, not with elaborate hairstyles or gold or pearls or expensive clothes, 10 but with good deeds, appropriate for women who profess to worship God. 11 A woman should learn in quietness and full submission. 12 I do not permit a woman to teach or to assume authority over a man; she must be quiet. 13 For Adam was formed first, then Eve.
1 Peter 3:2-5 2 when they see the purity and reverence of your lives. 3 Your beauty should not come from outward adornment, such as elaborate hairstyles and the wearing of gold jewelry or fine clothes. 4 Rather, it should be that of your inner self, the unfading beauty of a gentle and quiet spirit, which is of great worth in God’s sight. 5 For this is the way the holy women of the past who put their hope in God used to adorn themselves. They submitted themselves to their own husbands,
Genesis 24:65 65 and asked the servant, “Who is that man in the field coming to meet us?” “He is my master,” the servant answered. So she took her veil and covered herself.
Isaiah 47:2 - 2 Take millstones and grind flour; take off your veil. Lift up your skirts, bare your legs, and wade through the streams.
Deuteronomy 22:5 5 A woman must not wear men’s clothing, nor a man wear women’s clothing, for the LORD your God detests anyone who does this.
Matthew 5:27-28 27 “You have heard that it was said, ‘You shall not commit adultery.’28 But I tell you that anyone who looks at a woman lustfully has already committed adultery with her in his heart.
Mark 9:47 - And if your eye causes you to sin, gouge it out. It’s better to enter the Kingdom of God with only one eye than to have two eyes and be thrown into hell,
The point being made is that women should cover their heads and dress modestly - not just when praying but in general. This is also to protect man committing adultery with the eye and looking at women with respect and honour.