I lived in the area back in the 80's and still visit every other year or so. I have two suggestions.
Stay near an entrance either North or West and hit the park very early. Take coffee find a spot and enjoy the sunrise. It is also the best time to watch the antelope, elk and bears play. Depending where in the park you start. I found going to the Northern park first and work South. All the tourists are still sleeping and will be coming from the South.
Other thought is to stay in the park. Prices are not real bad with planning. Old Faithful and Grand Canyon Area. Just my two cents.