I'm fortunate to have relatives in Idaho so was able to visit Yellowstone for the second time two years ago. We rented a motorhome and drove down to the park the first week of June. There was snow in some areas of the park and the first night was a little cool but only the first night. We went hiking, fishing and had a great time.
The Tetons aren't far so absolutely make time to visit there. Beautiful "Big Sky" country!
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Photo from DubR's gallery.