Going over for my 7th TdF this year, and each year I have spent some portion of time in the Briancon area. More specifically, Monetier-les-Bains or La Salle des Alpes. From there, there are many great out-and-back rides within range. Col d'Izoard, Col du Granon, Col de Montgenevre, Col du Mont Cenis, Colle Sestriere, Col du Lauteret, Col du Telegraphe, Col du Galibier.
It's also about an hour's drive to Bourg d'Oisans, where you can pick up the Croix de Fer, Les Deux Alpes, Col du Glandon, l'Alpe d'Huez and others.
This whole stretch is mighty sleepy, especially this year since the TdF is avoiding the area all together. But if you're looking to get up, go for a long out-and-back, picking up a couple monument climbs, get back to the hotel, shower, nap, go to the hot spring baths, eat a HUGE dinner of mountain cuisine, go to sleep early, and do it again the next day, hard to beat.
Some other interesting options are the Burgundy (Bourgogne) area, which has a lot of great little 3 and 4 km Cotes. Same with the Rhone area (Orange, Avignon, Ige, etc.). More nightlife and stuff to do.
Finally, my teammates swear by the St. Lary Soulan area of the Pyrenees.
Bonne chance!