<<"Perhaps there are parts of the Swiss National Routes that will be
accessible in early/mid-June as well? I'm not sure which, though. ">>
There will be many sections of the National Routes open for good hiking in early June. They may not match what I believe you re seeking, however.
But, to further answer your specific question, check the height profiles for each of the National Routes at
I did not look at them all. (Of course, you can look at other routes as well.)
The Via Jacobi and the Alpine Panorama Trail caught my eye immediately.
The Via Jacobi is a pilgrimage route. It avoids high altitudes and minimizes difficult hiking trails, because of the range of people of all ages and health conditions who use it.
Here is part of the height profile, which has 5 sections.
Only one part ( after Einsiedeln) is above 1000 meters. That might be the only part with snow problems, and probably not even there, because of the way the route was developed, for pilgrims.
Parts of the Alpine Panorama Trail meet that condition, and so do parts of other routes.
I picked 1000 meters as a semi-arbitrary height. The snow line will probably be higher than that, but there may be shaded patches that are not snow free. The moderators and others can surely refine my guess at an altitude guideline for early June. It is not a sure-fire method to screen routes, and 1000 meters may not be the exact number to use, but it is one that I use as a first guideline when I'm walking at that time of year, or in the Fall.
The winter weather reports in Switzerland routinely report the altitude above which snow is falling, useful for skiers and for road travel.
The problem with that approach is, of course, that you will avoid essentially all of the High Alps. That seems to defeat the purpose of your hikes.
You can also factor in climate.
The BLS Südrampe rail-trail discussed earlier, and in the inventory of trails on this website somewhere , is on the dry, sunny south-facing wall of the Rhone Valley. Check stages 3 and 4 of The Walliser Sonnenweg, in this link:
I have walked it in early April, before it is officially open, and the winter debris was not yet removed. See attached images. (Note the red sign stating that the trail was not open that I missed.) I did it in error, and had no problems, but I'd not do it again, cause there were a lot of fallen rocks on the trail. I could have been blocked and had to backtrack a few km. In June, there will be a lot of walkers on it, and the trail will have been cleaned up.
I've reached the limit of my small ability to deal with High Alpine hikes at that time of year. Those with specific knowledge of local conditions will have to refine or refute my comments.
In the many years since 1980, when I've been in Switzerland in June, usually late June, I've had a lot of pleasant and scenic hiking or walking, but not up in the High Alps. We did find that some lifts/gondola cars, etc., were not open, which added to the problem of getting "up" in many places, and I had to check the SBB listings to see about opening dates for specific lifts. Those that are the principal access to mountain villages remain open all year; those that serve ski country/hiking country may not be.