Most people consider late August and early September to be the most consistent time of the year in the widest variety of locations. That said, in my experience, some of the best snow seems to fall in late July in the more northern regions. September is very spring like. It can feel like March or April in the northern hemisphere.
As a rule, I tend to start in the north and work south as the season progresses. The areas around Pucón and to the south are very stormy in July and August. This means rain in the valleys and poor visibility in the alpine. The exception to this is Bariloche, which tends to be in the rain shadow, so there are more sunny days between storms. Unfortunately, the wind tends to screw up the snow, but it has its moments of brilliance.
September is prime time for skiing volcanoes. The weather is more stable, but the snow levels are still low enough that there’s rarely any walking in dirt. There are still storms that roll through, but they tend to be one or two days, rather than one or two weeks. It ends up being a nice break and a good excuse to explore the valleys.
October brings even more sunny days and warmer temperatures. The resorts in the north will be closed, and the touring options are limited unless you are looking to bag a 6000-meter peak. In the south however, things are getting better and better. This is a slow time of year for tourism and you can have the mountains to yourself. The more southern volcanoes and Patagonia become ripe for adventure.
November is nearly summer, so to ski during this time means glacier-based trips in Patagonia or adventurous trips to Antarctica.