A captain, I see.Cool story for you. I was a Cavalry Officer in the US Army. On many occasions we wear a felt Stetson (go watch Apocalypse Now, or We Were Soldiers)... I was once told to remove my "cover" indoors... I told that ranking officer from a different division to go fuck himself and that if he wanted me to remove my Stetson he would have to remove it himself.
After quite a few "discussions"... the ranking officer appiligized to me!!! A Cavalry Stetson, in the US Army, as part of the uniform, is authorized to be worn indoors. I'll edit this with a picture.
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Here are the analogous Marine Corps regulations:
d. Headgear is normally removed indoors. Marines in a duty status and wearing side-arms or a pistol belt will remain covered indoors except when entering a space where a meal is in progress or religious services are being conducted. Headgear will be worn in Government vehicles, except when doing so would present a hazard to safe driving. Wear of headgear in privately owned vehicles is not required.
Growing up, my father was always covered when outdoors. He even wore a hat when he was in civvies. Habit dies hard for Marines. He was buried in his uniform. No cover, however.