Well Argentiere might be your best option. If grand Montets Lift is open and snow cover reaches to bottom, then you get 2000 meters vertical to play with which depending on your level might take 10mins to most of the day. Be warned though, the exit from the lift to get on the piste can be an experience to remember and the moguls get very BIG if it hasn't snowed for a while. If you are not hot skiers then Les Houches might be a better option for a day which is the other end of Chamonix.
Chamonix is not a linked up resort like the modern French resorts so you can't ski from one area to the next but for only one day each area has more than enough.
Check out Flyers-lodge run by Dennis Trott, a Brit, for cheap accomodation just down the valley from Chamonix. 20 mins by car into Chamonix and you pass by Les Houches on the way. He may even run his mini bus up there for clients.
He used to run a ski holiday company called Trott-Ski before he took up flying.
Still recommend a trip through the tunnel to Courmayeur for a days skiing though. Predominantly intermediate level. If its snowing in chamonix its probably sunny the other side of the Mont Blanc and vice versa.
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