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Jacquie Kennedy wore a similar hat called a "pill-box". 1960s. Made in USA but not a US military hat. Writing looks Greek to me, maybe Eastern European.
There’s been a ton of Slovak on here lately! This hat is Slovak, retailed in Cleveland, Ohio, which had a large concentration of Slovak people in the late 19th and early 20th century. The hat is probably c.1920. I’d bet a band member wore it. The maker’s mark is advertising custom stamps, badges, flags, etc.
Thank you interesting. The hat is very small
The extent of it being a "pillbox" is only that it is a Byzantine pillbox. It isn't related to the mid-century Jackie Kennedy connotations. This particular Jozeph Rozko hat is associated with traditional slavic dress. He was in business from the early 1900s through the 1920s.