I highly recommend this website to look for touring options in the alps:
The website allows searching for everything from day-hikes to multi-day trips from hut to hut. The detail page for each tour lists the months that the hike is recommended for. There are some that are good for May, such as this one: http://www.alpenvereinaktiv.com/en/tour/piz-bernina-4049m-ueber-biancograt/11637047/
The descriptions are typically in German, so Google Translate is your friend.
As far as what time of the year the huts are open, The German Alpenverein lists that on their website for the huts that they support: https://www.alpenverein.de/DAV-Services/Huettensuche/
You'll also want to check the websites of the huts, since they usually list when they open for the year.
In general I would recommend Austria over Switzerland for an alpine tour, since there is a wider selection of options. Last August I spent 4 days hiking in the Dachsteingebirge outside of Salzburg, which was stunning. That included staying at the Simony Hut and Adamek Hut, and climbing to the top of Hoher Dachstein.