Nurse caps are not hats.

nurse cap aliases into nurse hat
 
This should be reversed and “nurse cap” made the main name for the tag, since the item is a cap (usually shaped like a tiara as used in the show), not a hat (pillbox or any other). It can still imply “hat” tho, although generic “headwear” would be better (unless we want other non-hats like wimples and veils and coifs to also imply “hat”) but that’s a thing for another day.

Our definition of “hat” is just a bit broad. The alias may get reversed, but it’ll still imply hat.

Just as suggested.

The reason that happened in the first place was because “hat” had much higher usage at the time.