Matsumoto City Koike Town
I'm not knowledgeable in this department, but it seems reasonable to think this is probably a civil defense helmet since it gives a city/town/family name (in red).
Deputy Sergeant-Major [or Chief Petty Officer?]
I'm guessing at this because the last kanji [曹?] is rubbed and not clear ... and the kanji order is not correct ... it should be 曹長 "Sōchō".
I'm not 100% certain the name is Satō because the first kanji is rubbed and unclear.
The "Jōsō" might be shorthand for the naval rank 上等兵曹 Jōtōheisō / Chief Petty Officer.
Thanks for the help. The second one lid was the one stumping me. I thinking it was some type of local police lid or such.