You are right, May is not the best month for hiking or trekking in the high mountains. However, hiking and trekking from the second half of May is possible at some places and at lower altitudes.
We start running the trek to Merzbacher Lake from the middle of May. It is a 6-day trek in the eastern part of the country, at the foot of Khan Tengri (7010m). This trek has to be done with an experienced guide and good camping equipment.
Alternatively, it is possible to organise some 2 and 3-day hikes near Altyn Arashan and Jeti Ogus.
Cycling is also possible in May, but again not over the very high passes.
