It all depends on the snow year, but early-mid June is still going to have a lot of snow in the high country. Here’s a photo from June 6 of this past year at about 11500′ ft:
I was SKIING. There was enough snow at ‘only’ 10k ft or so to actually get some decent glides in. Higher and I was able to actually (attempt to. :)
Having said that, if you are looking for some areas to backpack in, the Lost Creek Wilderness is ab absolute gem and often opens up before most of Colorado.
A favorite LCW wilderness loop of mine
If you are not set on hiking the CT only, there can be some places to hike/backpack in for sure at that time of the year. Not that you *can not* hike most of the CT in early June, but not many people may necessarily want to cope with the logistics, navigation and physical exertion needed for typical conditions for that type of backpacking in Colorado.
Ping me if you want to know of some other lower-elevation ideas. I also have some Colorado Trail info here if you are curious