In Ryunosuke Akutagawa's short story "In a Grove", the monk talks about the girl on the horse wearing a type of hat called a 牟子 with the furigana むし. However, this word is not in the dictionary. But I have seen other readings of it as ぼうし。 Could this be the origin of the word 帽子? Or, perhaps 帽子 just overtook 牟子.
I did find the word 眸子 read ぼうし, but it means 瞳孔.
Is there any real connection between these words? Since, 眸子 is written similarly to 牟子 and has the same pronunciation as 帽子. And, both 帽子 and 牟子 basically mean "hat" or at least a type of hat.