There are number of hooded overcoats/greatcoats made for the military. They either have detachable hoods or built-in hoods because they are meant to protect from the elements when you don’t have the choice of going inside at will. I have one from the US Air Force which is a marvelous deep blue, very well made. It’s an officer’s greatcoat. Although this will infuriate Skald, it was a gift from my sister who picked it up for me at the base second-hand store. That having been said, visit the military surplus stores and second-hand stores near military bases and see what they have. I found this German army surplus coat online, which looks like it would fit your definition, except it may be too heavy unless you take out the button-in lining. That having been said, I think the military is your best bet to find this particular configuration. They don’t care a lot about fashion, and hoods are eminently practical, so they use them.
Enjoy,
Steven