With the court robe (noticing a pattern here?) went a cylindrical hat, worn by both the king and his soldiers. It was usually slightly wider at the top than the bottom, with straight (that is, not concave or convex) sides, sometimes canted slightly forward (rhomboidal in profile), and usually fluted on the sides. Judging by the scale, it was perhaps six to eight inches tall (15-20cm).
With the cavalry costume, probably indicating Medes in this period (the reign of Darius), was worn a tall domed hat. This hat was humped slightly forward and its bottom edge slanted down in back. It was tied around the bottom with a hatband that was knotted in back and trailed behind.
In Greek art, imperial soldiers are mostly shown wearing a peaked hat with long earflaps that could be worn loose or tied around the mouth, or, more rarely, over the top like a bomber hat, and a wider flap covering the back of the head. It is variously referred to by historians as a tiara or kidaris, but these may designate other types of headwear, or as a kyrbasia. Variants of this type at Persepolis may include the headdresses worn by the Armenians, Arians, Parthians, Sagartians, Soghdians, and most famously the Scythians. In period Greek art, the style normally depicted is the Scythian style, with short earflaps worn loose and a tall peak.
A last piece of headwear is a twisted band, worn without a hat by two soldiers (possibly Immortals) at Susa.
While the tiara may have originally been made from an animal skin, Herodotus describes the usual material for Persian hats as felt. I'm still looking into making my own hat. The tiara looks like it should be easy to make by modifying a Phrygian (or Smurf) hat pattern and adding ear and neck flaps. The cylindrical hat could perhaps be made from a long strip of felt for the sides and a circle for the top, possibly molding in the pleats and giving the top a sawtooth edge.
The domed hat will have to be blocked. I think it could be done by stretching a large felt hat over a ball or something - otherwise, you may have to go to a professional hatter. It must be at least eight and a half inches (22cm) from side to side and six inches (15cm) deep at the temple (and even deeper in the back. Modern round-crowned hats are just too small - if you cut the brim off a bowler, it'll just look like a skullcap. The domed hat should basically make you look like you have a lollipop on your head.
Next up: Trying to run an empire here.