Our church does not teach that women should wear head-coverings. On the other hand, we teach that women should not teach the Scripture or exercise ministerial authority over men. The two passages where these teachings come from both mention Adam and Eve. So why we do teach the one and not the other? The answer lies in what one of the passages includes.
1st Timothy 2:12-14 says that women should not teach the Bible to men, or wield authority over them. This is because God created Adam first, and Eve sinned first. Of course, these reasons mean little-to-nothing to a liberal, since they don't believe in a special creation of the human race. But Christ was the Son of God, therefore His opinions about creation and the fall of humanity into sin were true.
The apostle Paul didn't teach these things because he was a misogynist; that's progressive slander. Paul was Christ's apostle, so what he wrote was divinely inspired, and as a result without error. Paul also didn't write it because there were some sort of unspoken cultural problems going on in the church at Ephesus (the city in which Timothy, the recipient of this letter, lived and worked). There is no evidence that Christian women in Ephesus were being especially outspoken, or pushy, or rude. There is no evidence that Paul was thinking about local problems at all. That idea is a fiction, invented by those whose goal is to justify women's ordination.
Paul doesn't forbid women from preaching, just from preaching to men. Gifted Christian women can preach, just not to men. He doesn't say they can't share the word of God anywhere. Priscilla, alongside her husband Aquila, shared about God's word to Apollos. Paul doesn't forbid women from having authority,. They can be elected mayor of the town, or own a business. But they cannot have final authority over the men of the local church. That burden is reserved to qualified men, as we read in 1st Timothy 3. Paul's definition of church authority at the end of chapter 2 runs right into his teaching about elders in chapter 3.
But then turn over to 1st Corinthians 11. Paul combines spiritual truths with cultural principles, and it is this which differs the two passages. Paul says that Christ is the head of every man, meaning every Christian man, and specifically Christian husbands over their Christian wives. (Christian men everywhere are not the heads of my wife). Paul did not mean that men were the source of women (taking the Greek word kephale as "source" rather than "overseer"), since God is not the source of Christ (11:3b) -- and anyone who says God is the source of Christ is a heretic.
First, we remember that there is no teaching of head-covering anywhere prior in Scripture. The Old Testament never teaches this, nor does Christ. Second, the only time the Christian woman needed to cover her head was if she was praying or prophesying in the worship meeting (11:5). If she was not praying or prophesying in the worship meeting, it's implied she didn't need to cover her head. So the application is situational.
Third, Paul says it's disgraceful for women to have short hair. But this idea is also not found anywhere else in Scripture, unless, perhaps, it connects generally to the Old Testament's reinforcing of God-ordained gender distinctives. There is no Biblical evidence that short hair on women was or is absolutely disgraceful to God. Historical evidence suggests that it said certain things to the Roman society of the time, that the woman was morally loose. No Christian woman should ever want to be thought of as morally loose; that would indeed be a disgrace to her.
Paul says angels are watching our conduct (11:10). But angels are timeless beings, and they always watch our conduct. The fact that they watch us doesn't reveal what is right, it means they are watching to see if apply God's Word to our lives no matter what century we live in.
But head-covering cannot be a timeless rule just because it was based on the order of creation, because Paul cites a second theological absolute -- that Christian men and women are inter-dependent in the Lord (11:11-12). This truth has just as much authority as the first truth. If citing a theological truth supposedly demands head-coverings forever, then Christian inter-dependency in the Lord would inversely require no head-coverings forever.
Fourth, Paul cites natural consensus, which is a real thing but not a spiritual authority (11:14-15). It's in the nature of things for people to (usually) admire a woman's long hair. But that is what is customary among human beings. It says nothing about God's will or a rule's timelessness.
I hope you can see how different is 1st Corinthians 11:2-16 from 1st Timothy 2:12-14. The addition of other reasons -- a cultural view of short haircuts among women, the principle of respectability, and natural consensus about female beauty -- defined how a Christian woman applied the timeless truth of Adam and Eve's creation at that time, especially in worship-service behavior. Hopefully you also see that Paul never taught that Christian women needed to wear a head-covering all the time, or even throughout the entire meeting. They only had to wear one if or when they audibly prayed or prophesied.
1st Timothy 2 only states the order of creation and fall, and references nothing about public opinion, standard customs, or conduct symbolized by apparel.