Sep-Oct is a nice time to visit when the minimum temperature would be between 0 and 5 C. Weather would be fine, though there is a small chance of light rain
You will need to reach Leh and acclimatize yourself: full bed rest on
the first day and minimal activity on the second. You can proceed to
Hanle on the third, or better fourth day. It takes 7-8 hours
(effectively full day) to travel to Hanle from Leh. We have vehicles
plying once in a few days. You will thus need to make travel plans with
8-10 days available between Leh-Hanle-Leh.
He's also been kind enough to offer me to make arrangements for my stay and travel between Leh and Hanle. I think this might be a great way to start - from the highest observatory in the world!