The word "uncovered" in 1 Corinthians 11:5 &13 is ‘akatakaluptos’ which is an unusual word and doesn’t occur anywhere else in the NT. It can also refer to having hair hanging down loose, as opposed to its being fastened up on top of or behind the head. It is found in the Septuagint in Leviticus 13:45 where it refers to a person’s hair ‘hanging loose’. Corinth had a mix of cultures, Greek, Roman, Jew, Pagan and Christian etc. Each had different views on correct dress / behaviour for men and women. Women’s hair was a prime object ‘of male lust in the ancient Mediterranean world’ (Bartlett).
In the pagan cult of Dionysus practiced at Corinth, if women had their hair hanging loose (e.g. when taking part in pagan events) it indicated she was a prostitute for hire. Respectable Greek, Roman and Jewish women wore their hair fastened up (sometimes very elaborately with added ornaments if the woman was wealthy, which Paul and Peter rebuked in 1 Tim 2:9 and 1 Peter 3:3 – the time spent on their appearance should have been devoted to prayer, learning and good works. It is interesting that these hair styles must have been visible to the apostles and not covered up).
Again the members of the ecclesia needed to show by their appearance and actions that they did not have sexually immoral behaviour (Acts 15:28-29) and if they were married then they were being respectful to their partner. Verse 6 can be translated ‘for if a woman does not have her hair fastened up (‘katakalupto’ – again an uncommon word but linked to akatakaluptos which was used in Leviticus as hair hanging loose, or it can be translated as ‘cover’), she should be shorn, and since it is a disgrace to have her hair shorn or shaved, she should fasten it up (‘katakalupto’). Scripture is consistent with the repeated commands to avoid sexual immorality and this interpretation is consistent with the rest of scripture and the apostles’ letter to the early churches.
In verse 10, the Greek translates as ‘Because of this a woman ought to have authority (‘exousia’) over (‘epi’) her head. Translators have added words which are not found in the Greek text and this is possibly a cause of confusion for us when reading our modern translations. What they have added are ‘sign’ or ‘symbol of’. This dramatically changes the meaning of the text. The word ‘exousia’ is used for the person stated to have authority (over 100 uses in the NT) and ‘never in the Bible or other Greek literature is it used in the reverse sense of being in subjection to an authority rather than being the possessor of the authority’ (Bartlett). So the women in the Corinthian ecclesia have the authority over their own heads and they decide whether to have their hair to show they are chaste or sexually available to other men.
Paul writes in v14-15 “judge for yourselves: is it proper for a wife to pray to God with her head uncovered?’ Paul is answering matters they wrote to him about – and I think this is the start of his answer here: ‘For her hair is given to her for (‘anti’) a covering (‘peribolaion’). He ends with ‘If anyone thinks to be contentious, we do not have such (‘toioutos’) a custom, nor do the churches of God. Some translations have changed the word ‘such’ to ‘other’ but this is not how the Greek ‘toioutos’ is translated in any other situation (Strongs 5108).
God has given women hair as a covering and they should follow the example of our Lord and be faithful to God, Jesus and their marriage vows (both spiritually and fleshly). God does not look at our appearance but at our hearts (1 Sam 16:7).