Mice ate thru my friend's coop floor. So anything that will help to prevent that is worth it. We did 4x4 posts and 6 foot high hardware cloth. Of course, it doesn't come in 6 ft so had to piece together and had a 2x4 and stapled hardware cloth to that. Pretty sturdy run.
I think I'm going to go with keeping them in the enclosed run completely until I'm able to order in electric fencing for the outer chain link fencing; the hardware cloth is definitely going more than 2-3ft up the fence. 