Do headbands count as hats?
Nog was seen wearing a Ferengi-style headband (earband?) in several episodes. Not sure exactly which episodes these screenshots are from, though.
As a cadet:
As an ensign:
From Memory Alpha, here's a photo of a flag officer from the same era in the Kelvin timeline as in your original question wearing a cap in a different style:
I also found this example from ST:Enterprise. It's part of the desert uniform.
Here's a better image from Memory Alpha:
And a description of when it's worn:
Available for use with the excursion uniform and standard duty uniform was a black hat, embroidered with the specific registry number of the wearer's home assignment. Made of stretchable material, these hats were essentially one-size-fits-all "baseball caps." While these caps were generally worn during away missions, some officers wore theirs aboard ship.