I'm actually thinking mine isn't made for a hard hat. I've got a few carhartt coats and some definitely have monster hoods but this one doesn't seem to be that big.

I have that exact coat in grey. I bought it in maybe 2016 off the clearance rack at tractor supply for $40. One of the best coats I've ever owned for durability. I have sheep cows and pigs and wear it all the time. It gets abused and holds up to the abuse.