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"Has anyone hiked any portions of the Continental Divide Trail?"

I hiked the CDT in 2011. A pretty high snow year there too; I hiked southbound, starting in June, and recall much of the state of Montana being snow. Some snow still in the Winds in northern Wyoming but not too bad, and then snow falling again getting out of southern CO.

New Mexico, however, might be great in June. I certainly enjoyed hiking in there in Oct/Nov anyway. I particularly suggest the Gila River alternate route. Jonathan Ley maps show all of the alternates, and are available online for free.

Like most PCT thru-hikers, I too went through the JMT portion in June, and unless it's a very low snow year (?) I wouldn't suggest that if you're looking for a snow-free experience (!).

Trying to think of other good options, June might actually be late in the year for the Florida trail (?). The Pinhoti trail in the eastern U.S. is probably hikeable most of the year.

The other thing you could consider is coastal hikes. The Oregon Coast Trail, for example, is something unique, but offers various challenges too; I've hiked just a small chunk of it. WA state too offers coast hiking in the Olympics; my wife and I did that just a few years ago, in April as I recall.

Certainly the Grand Canyon is hikeable; I think it was March when I hiked down in there for a while. Some ice up top but then a lot warmer down in the canyon.

IMO, June offers a lot more possibilities very much depending on what you're comfortable doing and experienced enough to do safely. Hiking on consolidated spring snow isn't at all the same as hiking on/through newly fallen snow, especially given that you typically don't have to worry much about big new snow dumps falling on you. And flowing water is generally easy to obtain, so not need to melt snow to get water (I didn't carry a stove at all in the snow on the CDT). It's not for everyone, but it certainly can be done.