Well half our guys are old timers and like their salty worn stetson better than new ones. They're issued. The bosses like us all to have nice, clean, unfaded ones so they kind of suggest and push new ones on us. I tend to go hatless and even eschew the baseball hat these days with the fatigue uniform.
I have a toss on the back seat stetson for the odd time I should wear one, it's kind of beat but not quite a gabby hayes stetson. I have another couple still in the boxes. One has never been used, and the other I refer to my "funeral" stetson, which is worn with a dress blouse jacket and sam browne belt. I think the extra was a retiree's extra turn in/unused that was my size and was handed off to me because it was sitting on a shelf in a store room that someone wanted to clean out.
So I guess no, I would not catch flak for losing a stetson. Lose a portable radio or some other sn'd gadget and yeah you'll catch flak.
ETA- in 18+ years i've come across 3 lost badges, 1 at a range in the parking lot, 1 on a snowmobile trail, one handed to me by a guy who found it at the state fair. A trooper's stetson left at the local diner (which I dropped off to the local barracks as I knew the guy), one portable that one of my fellow employees left on the roof of his car and it flew off in the middle of an intersection. One of those, hey that looks just like one of our portables while turning at a light.
it was F'd up but was still accounted for.
Oh and one glock 23 magazine loaded in a town hall mens room.