Old Faithful the geyser and the old magnificent Inn built in the early 1900s is not that far from the western entrance. It was 1975 when I drove there around the first of June. It was still very cold with snow piled up to about 8' high on the side of some roads. The Ranger at the eastern entrance asked me if I had snow chains when I arrived. I didn't but he said I could probably make it ok without them in my 70 Mustang. We drove around the north side about 60 miles if memory serves to get to Old Faithful. I wanted to leave by the southern exit and go through Jackson Hole but the road from Old Faithful to the southern exit was not plowed during the winter and was still blocked by snow. So, I would have had to drive about 80 miles back around the north side and east side of the park to get to the southern exit. So, instead we left the western exit and drove about 135 miles total to Idaho Falls to spend the night. That was over 40 years ago so things may have changed a bit since then.
Oh, as we were driving south towards the Grand Canyon the next morning, I heard on the radio that the park was closed due to a snow storm! Early June!