El Chaltén is a very touristy mountain town in Santa Cruz, Argentina. It's a popular destination for hikers, and hitchhikers.
East towards Ruta 40 (basically everywhere - El Calafate, Los Antiguos, Rio Gallegos)
There's only one real exit, and you saw it when you entered. That's the way you've got to go. At the exit, there's a little bridge, after which there's a bit of a pull out, and it's a great place for hitching. However, in high season, there are a lot of people trying to hitch out, so expect a several hour wait. It might not be a bad idea to bring a book or something to do and just sit and wait while the people who got there before you get their rides because trying to hitch a ride five meters away from someone going in the same direction is kind of a waste of time/effort.
Note: if you're hitching here, and no one is stopping at all, there are probably people on the other side of the roundabout, about 50m before the bridge, who are having much better luck than you. Move there; it's not luck. It's not as good of a spot regarding the space for the cars to pull over, and you miss some of the traffic, but you'll be the first hitchhiker the drivers see.
If you can get a ride along the Ruta 40 that doesn't take you the last 90 miles into El Chaltén, you'll get dumped in the middle of nowhere, and if you don't have food, water, and a tent, it might not be the best idea. That said, in reality you probably won't have to wait very long for the next ride, during tourist season.
The winds here are mighty and the rain plentiful. Get as much roof and walls as you can. There are many houses in construction in El Chalten to shelter in against the wind, but it's probably best to get up early in case builders come. Also, there is an abandoned bus on the ride, one block into town after the Mirador restaurant. Finally, on the other side of the street (where you can hitch out) just before the bridge, there is a totally closed but empty building, where in the backside there is a small open room with 2 electrical plugs, and a good floor to sleep on. (All sleeping tips from March 2015)