The tricorne hat, for example, seen in many historical films, performs a very useful function during seasons of rain by channeling rainwater away from the wearer's face to the shoulder areas (often also covered with raincoats).
Therefore in my fantasy setting, inspired by many an old school setting and my own learnings and research about Philippine history, there would definitely be a tricorne equivalent -- perhaps for the conquering settlers of the land of San Lazario -- during the rainy season of the two-season land. Interestingly, the tricorne (according to Wikipedia, anyway) originated from Spanish soldiers evolving the use of the standard broad rim hat, and was brought to France in the 1660s during one of the wars.
And it would have both functional and dress versions of the same structure. As for the decorative colors, it's hard to say.
This also means that there would have to be places to hang hats in homes and in public places -- preferably areas that you would not mind getting wet.
Of course, the Enigmundian Inspancialo aren't the only source of hats. The locals have a variant of the salakot, a hat that has kept the sun and the rain out of the locals' faces for many years. It is usually made from rattan wood or perhaps woven from reeds. But there are wonderfully decorated versions worn by the headmen of the various barangays and tribes -- decorated with embossed silver and precious gems and coins and metals.
But the common folk should have a variety of designs that they will wear, and perhaps hang on walls when they enter. There must be some kind of cultural approach that deals with ownership, sharing, and borrowing hats, particularly for long trips.
And, perhaps even some magical hats in the future. But for now, it's a nice cultural touch to add.