As pointed out above, the seasons in Wyoming are a bit different than in North Carolina. I grew up in Wyoming and have been back for visits since. The standard Wyoming joke (but there's a lot of truth in it!) is that there are three seasons in Wyoming: July, August, and Winter. I've seen up to a foot of snow fall (although it soon melts) in July and August, too, at altitudes of 9,500 to 10,000 feet and higher.
Unless it's a very unusual year, there will be considerable winter snowpack still remaining during June. Streams will also be very high from the snow melt, making fording difficult and often dangerous. Some years, those conditions last until late July. I understand that was true last summer (as it was in the Cascades and Sierra).
Navigation is also an issue where the trails are still covered with 3-6 feet of snow. There is also the postholing issue where the snow has become soft, and you break through sinking to your crotch at every step..
Interestingly, in most Wyoming locations, 6,000 feet elevation is out in the desert or out on the plains, depending on your location. There is a low "timberline" below which trees won't grow, in most places 7500-8500 feet. Those areas will be fine in June, but I'm not sure what you're planning to hike there. If your Elk Mountain (there are a lot of them--every range has one or more!) is the one northwest of Laramie, visible just south of I-80, I wouldn't anticipate serious problems in June, although you'll still find plenty of snow in the thicker timber.
As with the others here, I have to ask how much experience you've had with the serious snow and ice conditions and the high-water stream crossings that you are apt to find at higher altitudes in June.
If you can postpone your trip for a month, by all means do so! Even early July will be difficult most years. If not, I would recommend the southwest.
Edited by OregonMouse (11/09/17 09:39 PM)
May your trails be crooked, winding, lonesome, dangerous, leading to the most amazing view--E. Abbey