I would not carry it inside public buildings such as; libraries, municipal, county and state buildings. When I use to live in Colorado Springs, CO I walked in the downtown library one day carrying a long knife on my belt. Security did not like that and they asked me to leave. As soon as I left, I put the knife in my backpack and came back inside and security didn't say anything.
I would first check with the local city attorney's office, county DA's office on the legality carrying a hatchet as open carry in public to be on the safe side.
Carrying a hatchet on your belt would be a benefit as a deterrent against scum bags that want to cause problems with you. Who wants to mess with a guy who is carrying a hatchet?