If you have not read Dan Wallace’s outstanding article, What is the Head Covering in 1 Corinthians 11 and Does It Apply to Us Today? I would encourage you to do so. You’ll find it in the “Prof’s Soapbox” section on our web site.
The material quoted below from, The Bible Knowledge Commentary, should also shed some light on this question.
Paul began (11:2-16) and ended (14:34-35) his discussion of Christian freedom as it pertained to worship with remarks directed primarily at the behavior of women in the Corinthian church. Some have questioned whether his comments in this section refer to the actual meeting of the church or to extra-church occasions in which a woman might pray or prophesy. The fact that Paul appealed to church practice elsewhere as a feature of his argument in this section (11:16) suggests that he was discussing church meetings. Modern distinctions between meetings of the church for worship and other meetings of Christians seem based more on expediency than biblical evidence.
11:2. The Corinthians had expressed to Paul, either in their letter or via their spokesmen (cf, 1:11; 16:17), that they remained devoted to Paul and to the teachings, the central doctrines of the faith, which he had communicated to them (cf. 11:23; 15:1, 3). For this Paul commended them: I praise you.
11:3. Paul no doubt appreciated the Corinthians’ goodwill toward him. But more importantly, he wanted to see behavior in keeping with a Christian’s calling. As a prelude to his exhortation, Paul characteristically laid down a theological basis. In this instance it concerned headship. The word head (kephaleµ) seems to express two things: subordination and origination. The former reflects the more usual Old Testament usage (e.g., Jud. 10:18), the latter that of Greek vernacular (e.g., Herodotus History 4. 91). The former is primary in this passage, but the latter may also be found (1 Cor. 11:8). The subordination of Christ to God is noted elsewhere in the letter (3:23; 15:28). His subordination to the Father is also true in His work as the “agent” of Creation (8:6; cf. Col. 1:15-20).
11:4. When a man prayed aloud publicly or exercised the gift of prophecy by declaring a revelation from God (cf. 12:10), he was to have his physical head uncovered so that he would not dishonor himself and his spiritual head, Christ (v. 3).
The alternate translation in the NIV margin, which interprets the man’s covering as long hair, is largely based on the view that verse 15 equated the covering with long hair. It is unlikely, however, that this was the point of verse 4 (cf. comments on v. 15).
11:5-6. It cannot be unequivocally asserted but the preponderance of evidence points toward the public head covering of women as a universal custom in the first century in both Jewish culture ([apocryphal] 3 Maccabees 4:6; Mishnah, Ketuboth 7. 6; Babylonian Talmud, Ketuboth 72a-b) and Greco-Roman culture (Plutarch Moralia 3. 232c; 4. 267b; Apuleius The Golden Ass 11. 10). The nature of the covering varied considerably (Ovid The Art of Love 3:135-65), but it was commonly a portion of the outer garment drawn up over the head like a hood.
It seems that the Corinthian slogan, “everything is permissible,” had been applied to meetings of the church as well, and the Corinthian women had expressed that principle by throwing off their distinguishing dress. More importantly they seem to have rejected the concept of subordination within the church (and perhaps in society) and with it any cultural symbol (e.g., a head-covering) which might have been attached to it. According to Paul, for a woman to throw off the covering was an act not of liberation but of degradation. She might as well shave her head, a sign of disgrace (Aristophanes Thesmophoriazysae 837). In doing so, she dishonors herself and her spiritual head, the man.
11:7-9. The man, on the other hand, was not to have his head covered because he was the image and glory of God. Paul based this conclusion on Genesis 1:26-27. A woman’s (a wife’s) glory and image was derived from (1 Cor. 11:8) and complementary to (v. 9) that of the man (her husband). Man, then, was God’s authoritative representative who found in woman a divinely made ally in fulfilling this role (Gen. 2:18-24). In this sense she as a wife is the glory of man, her husband. If a married woman abandoned this complementary role, she also abandoned her glory, and for Paul an uncovered woman’s head gave symbolic expression to that spirit.
11:10. Paul offered a third reason (the first reason was the divine order— God, Christ, man, woman, vv. 3-6; the second reason was Creation, vv. 7-9) why womanly insubordination in the church should not exist. Angels were spectators of the church (4:9; Eph. 3:10; 1 Tim. 5:21; cf. Ps. 103:20-21). For a woman to exercise her freedom to participate in the church without the head covering, the sign of her authority (exousia, a liberating term; cf. 1 Cor. 7:37; 8:9; 9:4-6, 12, 18), would be to bring the wisdom of God (Eph. 3:10) into disrepute.
Other (but less acceptable) explanations have been suggested for the words because of the “angels”: (a) evil angels lusted after the women in the Corinthian congregation; (b) angels are messengers, that is, pastors; (c) good angels learn from women; (d) good angels are an example of subordination; (e) good angels would be tempted by a woman’s insubordination.
11:11-12. Men and women together in mutual interdependence, complementing each other, bring glory to God (cf. 10:31). Neither should be independent or think themselves superior to the other. Woman’s subordination does not mean inferiority. Man is not superior in being to woman. Eve came from Adam, and each man born in the world comes from a woman’s womb (11:12). God created them both for each other (Gen. 1:27; 2:18).
11:13-15. Paul had based his previous reasoning for maintaining the head covering as a woman’s expression of her subordination on arguments rooted in special revelation. Now he turned to natural revelation (cf. Rom. 1:20) for a fourth argument in support of his recommendation. Mankind instinctively distinguished between the sexes in various ways, one of which was length of hair. Exceptions to this general practice were due either to necessity (e.g., Apuleius The Golden Ass 7. 6, “to escape in disguise”) or perversity (Diogenes Laertius, Lives 6. 65). No abstract length of hair was in mind so much as male and female differentiation. The Spartans, for example, favored shoulder-length hair for men (cf. Lucian, The Runaways 27) which they tied up for battle (Herodotus History 7. 208-9), and no one thought them effeminate.
Long hair was a woman’s glory because it gave visible expression to the differentiation of the sexes. This was Paul’s point in noting that long hair was given to her as a covering. Natural revelation confirmed the propriety of women wearing the physical covering (cf. Cicero On Duties 1. 28. 100). She has a natural covering, and should follow the custom of wearing a physical covering in a public meeting.
Some Bible students, however, say that the Greek anti, rendered “as” (i.e., “for” or “in anticipation of”) should be translated in its more normal sense of “instead of.” According to that view, a woman’s hair was given instead of a physical covering, for it in itself is a covering. In this view women should pray with long hair, not short hair. This view, however, does not explain the woman’s act of covering or uncovering her head, mentioned in 1 Corinthians 11:5-6.
11:16. Paul’s fifth argument for maintaining the status quo on head- coverings came from universal church practice. Paul was not trying to foist a new behavioral pattern on the Corinthians but simply to hold the line against self-indulgent individual excess in the name of freedom. As in the case of food offered to idols (8:1-11:1), Paul dealt with the immediate issue but also put his finger on the root of the problem, the Corinthian pursuit of self-interest which was unwilling to subordinate itself to the needs of others (cf. 10:24) or the glory of God (10:31). Throwing off the head- covering was an act of insubordination which discredited God.
Whether women today in church services should wear hats depends on whether the custom of head coverings in the first century is to be understood as a practice also intended for the present day. Many Bible students see that for today the principle of subordination (not the command to wear hats) is the key point in this passage. The intent of the custom of women wearing hats today, for fashion, seems far different from the purpose of the custom in the first century.