Head Covering Practices
Corinth was a Greek city, so we must assume that most worshipers followed the general Greek customs for worship.
"Among Greeks the habit was to offer worship with head uncovered. Reading for instance in Macrobius Ambosius' Saturnalia Convivia Bk. 1, 8, there the Greek manner of worship occurs with head uncovered." [Alford's Greek New Testament, Vol. 2, p. 564]
"The Greeks (both men and women) remained bareheaded in public prayer ..." [A. T. Robertson, Word Pictures in the New Testament, Vol. 4, p. 159]
"It is true that the Greek practice was to keep the head uncovered at their religious rites (as Grotius and Wetstein have remarked) ... " [W. J. Conybeare and J. S. Howson, The Life and Epistles of St. Paul, The S. S. Scranton Company, 1920, p. 445]
"Greek women did not wear hats, but they did wear their hair in a variety of different styles, sometimes adding a ribbon or a scarf. They also wore jewellery especially earrings and necklaces." [Winged Sandals, Australian Broadcasting Company, 2003]
"The mysteries inscription of Andania (Ditt. Syll.3, 736), which gives an exact description of women taking part in the procession, makes no mention of the veil. Indeed, the cultic order of Lycosura seems to forbid it. Empresses and goddesses ... are portrayed without veils ..." [Oepke in Kittel Theological Dictionary of the New Testament 1965, vol. 3 p.562].
At the time of the New Testament, Corinth was under Roman rule. It's customs and practices would be evident in the city.
"Our heads are shrouded before the altar with a Phrygian vestment" [Virgil (a Roman poet c. 70 B.C.), The Aeneid, Bk. 3, p. 545]
". . . and when now thou raisest altars and payest vows on the shore, veil thy hair with covering of purple robe, that in the worship of the gods no hostile face may intrude amid the holy fires and mar the omens" [Virgil, Aeneis Bk. 3, p.403-409].
"Why is it that when they worship the gods, they cover their heads, but when they meet any of their fellow-men worthy of honour, if they happen to have the toga over the head, they uncover?" (Plutarch, Moralia, The Roman Questions 10)
"It is no piety to show oneself often with covered head, turning towards a stone and approaching every altar, none to prostrate upon the ground and to spread open the palms before shrines of the gods . . ." (Lucretius de Rerum Natura 5.1198-1201).
"It was in accordance with the traditional usages, then, that Camillus, after making his prayer and drawing his garment down over his head, wished to turn his back; . . ." (Dionysius of Halicarnassus The Roman Antiquities 12.16.4).
Several of the members of the church in Corinth came from Judaism, so its influence would be seen as well.
"The meeting of the congregations in the ancient synagogues may be easily realized, if due allowance be made for the change of costume, by those who have seen the Jews at their worship in the large towns of Modern Europe. On their entrance into the building, the four-cournered Tallith was first placed like a veil over the head, or like a scarf over the shoulders." [W. J. Conybeare and J. S. Howson, The Life and Epistles of St. Paul, The S. S. Scranton Company, 1920, p. 154]
"Let not the Wise Men, nor the scholars of the Wise Men, pray unless they be covered" [Maimonides, quoted by Lenski, The Interpretation of St. Paul's First and Second Epistle to the Corinthians, 1946, 435].
"Men sometimes cover their heads and sometimes not; but woman's hair is always covered, and children are always bareheaded." [Talmud, c. 200-500 A.D., Nedarim, p. 30b]
"What is the Jewish law? Let not a woman go with head uncovered. This is founded in the Law, for it is said (of the suspected wife) 'the priest shall uncover her head' Numbers 5:18." [Talmud, c. 200-500 A.D., Chetubb, fol. 72.1]
"Rav Huna the son of Rabbi Joshua never walked four cubits with his head uncovered. He said 'because the Divine Presence is always over my head.'" (Talmud, c. 200-500 A.D., Kiddushin, 32a)
"Artistic representation, such as Egyptian and Babylonian tablets or the synagogue at Dura Europos, generally depict Israelites (and later Jews) without head coverings ... According to the Talmud (Ned. 30b), it was optional and a matter of custom for men to cover their heads." [Encyclopedia Judaica, Vol. 8, p. 16]
"An example of a custom that assumed the status of law ... is the requirement that Jewish men always keep their heads covered. This began as the personal practice of some of the scholars in Babylonia, who considered covering the head a sign of humility. It spread from them to the Jews of Spain and from there to other communities in Europe during the Middle Ages." [Roy A. Rosenbery, The Concise Guide to Judaism, p. 125]
"Among the Jews, so usual is it for their women to have the head veiled, that they may thereby be recognized." [Tertullian, c. 160-225 A.D., De Corona, ch. 4]
"The Midrash contrasts the attitude of Moses in hiding his face before the Shekinah at the burning bush (Ex. iii. 6) with that of Nadab and Abihu, who looked on with uncovered heads (Ex. xxiv. 9, 10): the one showing reverence and awe; the other, insolence (Ex. R. 3). The proper attitude, therefore, of one called upon to pronounce the name of God in prayer, the "Sheliah Zibbur," is to be wrapped in the mantle or tallit (R. H. 17b; Ber. 51a; Yer. Ber. vii. 11d ). Accordingly, a man with uncovered head is, like one in rags and half-covered, forbidden to recite the Shema'—or, at least, to officiate as Reader or to read aloud from the Torah or to recite the priestly benediction—he not being in a position to pronounce the name of God with proper dignity (Mas. Soferim xiv. 15; compare ed. Joel Müller, p. 199; Azulai, Responsa "Hayyim Sha'al," ii. 35)." ["Bareheadedness," JewishEncyclopedia.com]