For some reason headcoverings seem to be a controversial issue within the Church. Do please note that the author of our Divine Liturgy, Saint John Chrysostom, instructs that we adhere to this standard during Divine Liturgy. Since he is indeed the author/editor of the Liturgy, we might want to take what he says about the Liturgy seriously.
Clement of Alexandria ca. 150-215
Woman and man are to go to church decently attired, with natural step, embracing silence, possessing unfeigned love, pure in body, pure in heart, fit to pray to God. Let the woman observe this, further. Let her be entirely covered, unless she happen to be at home. For that style of dress is grave, and protects from being gazed at. And she will never fall, who puts before her eyes modesty, and her shawl; nor will she invite another to fall into sin by uncovering her face. For this is the wish of the Word, since it is becoming for her to pray veiled. (The Instructor 3.11)
Tertullian ca. 160-220
Arabia’s heathen females will be your judges, who cover not only the head, but the face also… And the modesty of heathen discipline, indeed, is more simple, and, so to say, more barbaric. To us the Lord has, even by revelations, measured the space for the veil to extend over. For a certain sister of ours was thus addressed by an angel, beating her neck, as if in applause:
Elegant neck, and deservedly bare! It is well for you to unveil yourself from the head right down to the loins, lest withal this freedom of your neck profit you not! And, of course, what you have said to one you have said to all. But how severe a chastisement will they likewise deserve, who, amid (the recital of) the Psalms, and at any mention of (the name of) God, continue uncovered… (On the Veiling of Virgins 17)
St. Hippolytus of Rome ca. 170-235
When the teacher finishes his instruction, the catechumens shall pray by themselves, apart from the believers. And [all] women, whether believers or catechumens, shall stand for their prayers by themselves in a separate part of the church. And when [the catechumens] finish their prayers, they must not give the kiss of peace, for their kiss is not yet pure. Only believers shall salute one another, but men with men and women with women; a man shall not salute a woman.
And let all the women have their heads covered with an opaque cloth, not with a veil of thin linen, for this is not a true covering. (Apostolic Tradition Part II.18)
St. John Chrysostom ca. 347-407
For this cause ought the woman to have a sign of authority on her head.
For this cause: what cause, tell me?
For all these which have been mentioned, says he; or rather not for these only, but also
because of the angels.
For although thou despise your husband, says he,
yet reverence the angels.
It follows that being covered is a mark of subjection and authority. For it induces her to look down and be ashamed and preserve entire her proper virtue. For the virtue and honor of the governed is to abide in his obedience.
Again: the man is not compelled to do this; for he is the image of his Lord: but the woman is; and that reasonably. Consider then the excess of the transgression when being honored with so high a prerogative, you put yourself to shame, seizing the woman’s dress. And you do the same as if having received a diadem, you should cast the diadem from your head, and instead of it take a slave’s garment. (Homily 26 on First Corinthians)
Blessed Augustine of Hippo ca. 354-430
[W]hat the apostle meant to signify is plain, and in so far figuratively and mystically, because he was speaking of covering the head of the woman, which will remain mere empty words, unless referred to some hidden sacrament. (On the Trinity Bk. 12.7.11)
St. Paulinus of Nola ca. 354-431
Let them realize why Paul ordered their heads to be clothed with a more abundant covering: it is because of the angels, that is the angels who are ready to seduce them and whom the saints will condemn…A woman ought to cover her head especially in prayer and prophecy. Then she becomes pregnant with the Spirit, and accordingly rouses the hatred of the tempter all the more when she leaves behind the boundaries of her womanly weakness, and aspires to human perfection…So because a woman becomes spiritually pregnant,…she has a power over her head, so that the wiles and snares of the enemy may not confront her. (Letter 23 to Severus)