So, according to scripture, the women in your church are required to wear hats while attending services if they have short hair and are forbidden to wear pants, right? Likewise, they are forbidden to braid their hair, wear any gold or pearl jewelry? Are they also forbidden to speak in church?
...Every man who prays or prophesies with his head covered dishonors his head, but every wife who prays or prophesies with her head uncovered dishonors her head, since it is the same as if her head were shaven.
...But if a woman has long hair, it is her glory? For her hair is given to her for a covering.
...Likewise also that women should adorn themselves in respectable apparel, with modesty and self-control, not with braided hair and gold or pearls or costly attire...
...The women should keep silent in the churches. For they are not permitted to speak, but should be in submission, as the Law also says. If there is anything they desire to learn, let them ask their husbands at home. For it is shameful for a woman to speak in church.
I find nothing in scripture that says a man must remove his hat except while praying or prophesying.