In Greece, outside Old Calendarist parishes, covering is not a thing, and it hasn't been for a long time, it's not a matter of 'modernity' or 'feminism', or whatever. (My grandmother, who lived in a village all her life, wore a headscarf daily - except at church on Sundays.)
I can only assume that the headscarf was something that women, especially rural ones, wore to keep their hair clean and tidy while working, and as a work tool, it was not considered appropriate when dressing up for church. Urban women, who didn't wear scarves daily, naturally didn't take to wearing them at church; not sure if hats were ever popular for the occasion, but their time was over by the 1960s anyway. The only exception was widows, who didn't take off their black scarves unless they remarried, a custom that lasted somewhat longer. These days it's rare to see one of those, as they are of very advanced age.
In my Antiochian parish in the UK, only two ladies cover consistently; one of them is Serbian and continues the tradition she grew up with, the other is elderly and opts for a knit hat for about half of the year. A couple more only cover when they go up for Communion, and another, when she's in the choir. We're a multinational gathering, so no one bats an eyelid either way.