Thanks for the question.
This is sometimes a difficult part of Paul to understand because there are many different facets of his arguments. It is actually very significant in light of today's issues as well. But let me try to break it down into the basic point.
Paul views long hair as a defining element in the distinction between men and women (in Roman, Greek, and Jewish culture men wore short or relatively short hair), according to custom, but really as a symbolic reification of the headship of man over women. It is patriarchal. Women are under the protection and headship of man (the hair is like a "tent"), but conversely, men are also tied inextricably to women inasmuch as they are born of women. This is an ordering that comes from Genesis 2, and creates both a hierarchy and equality, a paradox of sorts, but nonetheless a distinction of the roles of the sexes, which really serves to point us toward the relationship of man under God. All mankind is under the headship of Christ, and though we are not equal to God, He has condescended to us to bring us into His Own Life. The veiling of women, which is really just the clothing form of the hair distinction (which reified the metaphysical distinction) serves as a reminder to us all of our place in God's Kingdom.
So, today you can see the culture seeking to dismantle all signs and evidence of distinction between the sexes, their roles, bodies and metaphysical dimension—-never mind the notion of a patriarchy ordering us under the headship of Christ.
For Paul, customs are significant because they can point to invisible truths—and Paul doesn't just like customs for the sake of customs; he is the first one to get rid of anything that might interfere with pure and true worship. So for Paul to advocate for this act of women (having heads covered) really means he thinks it's important for greater theological symbolism. That's why in traditional Catholicism, women still use the veil for worship at Mass. There could be an argument made that if the Catholic Church did a better job of holding on to traditions that underline good theology, then perhaps the culture would not disintegrate so rapidly.
More than any institution, the Catholic Church is the most important bulwark against the corruption of morals, philosophy, theology, and metaphysics. But in these last decades we have undone ourselves. Let's pray we can do better.
Feminists point to patriarchy as a way to indict men as oppressors, rather than protectors, of women. They are really indicting God, a continuation of the rebellion in the garden, for which we have all suffered. Only when we are all under the mantle of Christ, will we not only be protected, but truly free.
- Have you ever heard someone pray to be covered by the mantle of Our Lady?
That is the ultimate clarifying point—she is under the total protection of her Son, and she has so much room (under this tent) to move because she is truly free, that she invites us into the space under her mantle. That is the truest veil, not oppression, but freedom, joy, peace, love and sharing. How happy am I to be "covered" by her tent!
- Do you see how much our Mother can show us the fulfillment of Gods design?
Don't buy into the culture which has been duped by satan to rebel against the beautiful design, harmony, and plan of God to make us all safe and joyful under his mantle. Embrace your femininity and rejoice in how God made you. We may not have cultural markers that underscore this important truth, but we can still live under the mantle of God's grace, through our Lady, and embracing our total personhood in Christ.