I have a very large desert, approximately 15 000 km across from west to east (the north-south distance is around 4 000 km, although it's at a bit of an angle). The desert is bounded at one edge by a large mountain range running north to south which casts the rain shadow which gets the desert started. This desert is intended to be extremely inhospitable, with the bulk of it receiving no rainfall and having no sources of water available for human travellers.
About 9 000 km into the desert (from the side with the initial mountain range) is a secondary mountain range, also running north to south. The desert then continues another 4 000 km.
What I am wondering specifically is, is it technically plausible for the mountain range cutting through the desert to have accumulated snow caps? I would like for that mountain range to serve as a way through the desert (north-south), albeit a very difficult one, by virtue of having a small amount of moisture present rather than none at all.