If you are near the Grand Canyon, don't pass up an opportunity to visit Monument Valley (Navajo Tribal Land) on the border of Arizona and Utah. Absolutely gorgeous, then consider traveling north to Arches MP. If you want, you can circle around to Bryce and Zion NPs.
Heed the posts about the weather. It can snow it Yellowstone any month, and one June we were snowed in for a day. Another visit and another June, temps dropped to freezing. Consider seeing parks in the southwest and gradually work north as the weather improves. Have fun.