Take a look at the "Walk In The Park - event report" thread below in the Wyoming forum. There is a good description of our reception over the Independence Day week.
It would not hurt to stop by the town hall and ask to talk to Chief Todd
Smith or Lt. Bob Gilliam, you can reference me, Bob Culver and the Independence Day Open Carry. Just tell him you are new in town and plan to open carry at times and see what he recommends. I imagine it will be very much like in the open carry event write-up.
If you go into private businesses, there might occasionally be a comment, but I very much doubt any serious bad times. If anyone objects and asks you to leave just explain that it is unfortunate they do not want your business and walk out. Only if you persist and remain after being asked to leave can they claim any kind of trespass violation.
Be careful in bars and restaurants with alcohol; you might ask Chief Smith about this, I never looked into it.
Be sure to go south of town and visit the Jackson Hole Gun Club <http://jacksonholegunclub.net/>