The Church of God cites Paul to argue that women should wear a towel on their heads. 

[1 Corinthians 11:1~2] Be imitators of me, just as I also am of Christ. Now I praise you because you remember me in everything and hold firmly to the traditions, just as I delivered them to you.

Paul told the church in Corinth that women should wear a head covering when praying, but he made this claim because of Jewish tradition, not because of God's teaching. Old Testament law-abiding Jews had a tradition of wearing head coverings, just as women in Islam do today, and Paul was advising them to do so in order to maintain order in the church. 

[1 Corinthians 11:7~10] A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. For man did not come from woman, but woman from man; neither was man created for woman, but woman for man. For this reason, and because of the angels, the woman ought to have a sign of authority on her head.

In the Bible, men were not born from women, but women were born from men.  When Paul advised women to wear a towel on their heads, it was a sign that women should show that they are under the authority of men. Paul's exhortation for women to wear a towel on their heads was because the Corinthian church was being disrupted by women. Many of the women who had entered the Corinthian church were adding to the disruption by teaching and prophesying at will, ignoring the authority of men in the church. Paul wanted to restore order to the church by correcting the men's authority by having the women wear towels on their heads.

[1 Corinthians 14:33~35] -For God is not a God of disorder but of peace. As in all the congregations of the saints, women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the Law says. If they want to inquire about something, they should ask their own husbands at home; for it is disgraceful for a woman to speak in the church.

So when Paul exhorted women to wear a towel on their heads, he was following Jewish tradition, not speaking as a rule from God. Women should be silent in church, but God's church is centered on and driven by women.