1 Corinthians 11:4-8

1. Every man praying or prophesying, having his head covered, dishonoureth his head. 5. But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. 6For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered. 7. For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man. 8For the man is not of the woman; but the woman of the man.

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I sincerely doubt that NOT wearing a head covering makes any woman a bad woman. But, if you're going to take your faith seriously and strive towards being pious in your thoughts then adhering to traditions is important for you. There is no shame in being a woman and submission does not mean inferior or incapable. Also, traditions are not "patriarchal social constructs."

The bible says this about traditions

1 Corinthians 11:1-2 1. Be ye followers of me, even as I also am of Christ. 2, Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you.

I won't be a moralist and lecture others what they should be doing. I just want to say that other women should consider covering their heads