In the famous Alexander Mosaic found in Pompeii, Darius is wearing some form of head covering, it appears to just be cloth wrapped around his head. What is the hat called? I can't find any information on it. Was this common combat attire, or just artistic?
From this website http://www.iranicaonline.org/articles/clothing-ii , in a section concerning historical clothing it goes into length on several styles of headgear. This style falls under the designation 'tiara'.
In any event, the tiara had a top like a hood, often lined inside with luxurious animal fur. Ordinarily it was worn flat, either pressed down in front to form three knobs or falling in folds on either side. Only the great king had the right to wear his tiara (kyrbasía) “upright,” that is, with the top erect, presumably held by inner retainers (Xenophon, Anabasis 2.5.23; Arrian, Anabasis 3.25.3; Plutarch, Artaxerxes 26, 28; idem, Themistocles 29). (emphasis mine)
from the mosaic:
and a normal 'tiara' from the above mentioned web page shown in 'flat' style:
Remarkably similar to the Papal Tiara, although I don't know if this is mere coincidence. Feb 27, 2019 at 15:16
It looks like a padded coif:
OT aside: you might also find this Youtube video from Lindeybeige on mail coifs interesting.
@user2448131's answer seems better, however, I wouldn't be surprised if the coif were derived from the Persian tiara.– tox123Sep 17, 2016 at 1:45