You could fly to Santiago and ski Portillo, then Valle Nevado/La Parva/El Colorado (which are connected by chairlifts) and then go 250 miles south to Nevados de Chillan (drive, take a train or fly to Concepcion - skip the bus) they usually have a very reliable snow record and it's a volcano.
Further south is Pucon, another beautiful Volcano area, really quite place but the weather there is not very reliable... it may snow tons and then rain on top of it.
In Argentina I would recommend Cerro Catedral (near Bariloche) and one weekend in Chapelco, (2h30min drive from Bariloche, near San Martin de Los Andes) not huge or very steep resort, but generally uncrowded and with nice views.
There is also Las Lenas and Cerro Castor. I've never been to none of these two resorts, Las Lenas (close to Mendoza) seems to have a great terrain but is expensive and has uncertain snow records, and I also heard that the best terrain is usually open only during 25-30% of the season.
Cerro Castor is the southernmost ski resort in the world, near Ushuaia. It's not big but has good vertical and good snow till october.
In my opinion, August is the best month to ski South America, since most the snow falls during June, July and August. In September sometimes things can deteriorate pretty fast.