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I'm seeing Carmen Sandiego in it for some reason, but the modern version (from 2014) has a film noir detective in the same spot (the application icon is still the lady in the hat): https://youtu.be/h-_UNm_gycU?t=94

For me it's the spy from the wep chips challenge game

It’s from the ”Spy vs. Spy” comic strip.

Sibling already mentions this, but the wording of the article almost directly but not quite says it's "Spy vs. Spy" based. I'm not aware of an official link to document this but

https://www.google.com/search?q=spy+vs+spy&udm=2

An image search will show you plenty copies of the inspiration for this hat.