Happened to come across this on my recommended's as I was scrolling away.
I'd like to make an addition to this as a Japanese major!
So mushi no tareginu (č«ć®åć蔣) isn't actually the hat, it's what the veil is called. The kanji used to describe this veil is literally insect's dripping silk.
The actual hat is uchikatsugi. It kind of comes from the Chinese Mili (which in turn was derived from (presumably) Mediterranean cultures).
Uchikatsugi was used my noblewomen most often, usually to hide their appearance when out in public and even by dancers. Uchikatsugi was not commonly used by men. (It was used, just not with the same commonality).
https://www.seiyaku.com/customs/veil/veil-japan.html
Scaramouche, a soldier, a warrior did not actually wear an uchikatsugi (at least not likely). This probably is assumed because of the veil on his hat, but veils were added to other hats as well as a way to guard against mosquitos and the like.
Scaramouche likely wore a Jingasa é£ē¬ ć These are hats with a similar wideness of the brim but were made to be protective and most importantly they were lacquered. This allowed for more durability and use when travelling and were worn by samurai and other warriors.
(Top example is an 18th century Edo period, iron and brass Jingasa; bottom example is a Ichimonji Hira-jingasa from the end of the Edo period)
Jingasa (a hat) can have a mushi no tareginu (veil)
This website has a pretty thorough breakdown of (most but not all) types of hats in historical Japan.
The Met museum has a lovely example of an 18th century Jingasa (that actually resembles Scaramouche's)
https://www.metmuseum.org/art/collection/search/22096
All this said, as a game character, his design is very glitzy. Soldiers weren't particularly likely to have their headwear draped with the finery and designs Scaramouche has when fighting. But the accessories and design definitely show his prowess.