May is the "sacred" month for morris dancers- traditionally, it's THE month for morris dancing in most traditions.

I went to JC's ABC Tunefinder and typed May into the search engine. There are lots of tunes with May in the title; not all I recognize as morris tunes.
The ABC library of morris tunes lists some tunes with May in the title:
17th of May
29th of May

Not much else in the way of current tunes being used for morris these days.
I would be very interested to see/hear the tunes you've found, Ian P!

Allison