Petasos (Sun Hat)
The petasos was the name for a basic straw or fulled wool hat that had been in use since Archaic Greece. These were adopted by the army as sunhats to be worn when off duty or performing daily chores, and remained in use well into the medieval era. Although they take various forms in art, one such hat has actually been found; it is housed in the Pitt Rivers Museum and dates to the 5th century AD, and is possibly the oldest Straw hat in the world.
Pileus (Freedman's Hat)
The pileus was a brimless cap that began to be worn in ancient Greece in the Archaic and Classical period, and was later adopted by the Romans. Initially conical in shape, hemispherical versions were worn by the Romans, sometimes with very short brims. The hat was considered a mark of liberty, and manumitted slaves were given such a hat in undyed or bleached wool. They could be constructed of fulled wool, or made using naalbinding. Such hats are markedly similar to traditional fulled wool hats still worn in remote parts of Albania. Eventually, somewhat more elaborate forms of this hemispherical hat became known as the kamelaukion, and became an element of court dress in the "Byzantine" period.
The phrygian cap is a tall hat somewhat resembling the classic "santa claus" hat, but without the fuzzy ball on the tip. These hats were presumably associated with the Phrygians in Asia minor (although were of Scythian origin), and became widespread in the Roman Army from about the 2nd century BC through the 7th century AD. Phrygian caps could be constructed of a variety of materials in combination, however Roman ones typically seemed to be made of fulled or twill wool or cashmere, or also possibly linen, cotton, or silk. They also could be constructed by placing a fabric material over a fur lining such as sheepskin or squirrel. Their construction also came in two forms: either they were built as a single piece with a hemispherical shape that came to a bulbous point which naturally folded over, or they could be constructed with two pieces and sewn up the middle. Some Romans wore these hats in the steppe fashion, where the hat wrapped around the sides of the head and was tied underneath the chin, or alternatively had the sidewall flaps tied back behind the head to expose the cheeks and ears. Roman soldiers would typically wear these hats when off duty or performing daily chores, but they were also often used by unarmored skirmishers or light infantry and cavalrymen usually found in the auxilia.
Soul of the Warrior - Owned by the Centurio of Legio VI, this site will cover many of your clothing-based needs.
Medievaldesign - Offers a good straw Petasos, a Pileus, Phrygian cap, and a Cucullus.
Fabrica Cacti - Makes hats in felted wool, twill, or naalbinding.
LuaMedia - Sells phrygian caps in various colors.
Helgi's True History Shop - Sells a felt Petasos, Pileus, and Phrygian caps.