Early Season Southwestern Colorado

Even southwest Colorado will have snow and ice in the upper mountains, though I could usually start backpacking around the CDT/CT juntion east of Silverton late June. Rule out Purgatory Creek and the Animas – scenic but not what you are looking for IMO.

Also some of the lower elevations have experienced fire warnings already, so maybe some of the lower elevation areas east of Durango, trails near Vallecito Ck, Florida Ck, and Los Pinos Ck., will take you into the Wemuniche wilderness, though you will need to circle Vallecito Reservoir with all it's RVs.

If you could widen your criteria out a bit from S.W. Colorado to include the Sangres then you might have a better shot regarding conditions. Currently the Sangres are very dry relative to other ranges in Colorado and on an absolute basis.

South Colony Lakes might be accessible by then, Willow Lake on the west side has a better shot, high mountain lakes though are usually still frozen at the time of year in part or whole.

The flattops are lower elevation, might be melted out better?? But not in S.W. Colorado.

Or what about New Mexico?

if you go a bit north you could do something in the Black Ridge Wilderness outside of Grand Junction. It's more redrock desert and canyons. Jones, Mee and Knowles canyons are all awesome, and Rattlesnake canyon has the highest concentration of arches outside of arches nat. park. They all empty into the Colorado and I'm pretty sure that any of them can be linked for a loop. Usually very few people, if any.