You'll find Verizon in some areas of the park and one is not far from Fishing Bridge. No need to drive outside the park. When you arrive, just ask at the Visitor Center for help in locating the best possible areas. Also ask for a park telephone number that your family can call - in an emergency only - and that someone will contact you. It's easy to do.
As far as cold temps - in June in the 20's would be rare but you could have around 35-40 deg. so bring extra blankets! It does warm up fast though during the day. .. and yes, you 'could' have light snow but it would melt fast during the day. You'll have a wonderful time!!