June can be impressive as the peaks will have the snow on them, but you might get a fresh load dumped on the mountain pass closing it for a couple of days. June / July can have the TdF closing roads for the race, though afterwards the surface should be good.
The southern part I'd say is great in September, away from the heat of August, which can trigger off TdF stage stopping style thunderstorms, but the chance of snow, except on the mountain tops is unlikely.