HI Bob, This year the south half of the range is in deeper snow than the north. I am headed out to ski on Wyoming pk this weekend so I'm not sure what you will get the last week of June but at the least snowfields to post hole through. Of the entire range the Scenic Trail is a mostly unmaintained route, used in winter for snow machines in summer its an elk trail, a use trail or no trail so don't expect a clearly marked path and then you will be fine. Of the North half, Camping and water are hard to come by on the N section and the "sure water" at Gunsite pass was a very muddy mess you don't want to use that one. There were a lot of blowdowns coming into the little grey's river area last year. There should be more water this year however every water/camp area suggested by the forest service was unusable last year so camp as soon as you find a decent area./ Around the numbered creeks there is no trail and nav is easiest with a good old-fashioned compass. We learned to carry more water than we do in the winds so we could take advantage of nice campsites and skip steep scrambles into lesser water sources. We had a standoff with a Moose at 4th creek and had to brushwhack around.
The ground is hummocky everywhere so bring a thick pad, there was a wolf pack that followed us for days and lots of bear scat so be prepared for that. Finally, the FS designated the end of the trail further N than it is and the trail along willow creek becomes a bog so we exited out Adam's creek and with alter exploration determined this is the best route. The trailhead is found here (turn off 191 at the fiberglass moose). It probably goes without saying but your last day is deep in willows full of moose and sometimes bear so have water shoes and be prepared.
I hope that helps!