Your observation is correct that Jewish people do cover their heads not just when they pray, but during the whole synagogue service. Orthodox Jews even cover their heads throughout the day. This shows that Paul’s instructions in I Corinthians 11:2-16 are not coming from his Jewish/rabbinic background, but that he is laying down other rules and regulations for believers of the body of the Messiah. So, if you take the passage literally (as I do), the men should have their heads uncovered, whereas the women should have their heads covered. When I speak in my own Messianic congregation where I am a member, I do not cover my head. My wife, on the other hand, does cover her head when she attends any church meeting, since we both take the text very literally.
