Do they have different enough hours that you could try out the new place when the old place is closed?

Or just go the other direction so you don't have to walk by the old place to get to the new place. You deserve to feel comfortable when getting your hair cut.

Another way to think of it would be supporting with your money someone that politically you want to support. That might help you get over some of the uncomfortability of switching. And you can always go back to the old place if you end up not liking the new place after all.