El Calafate is a city in Santa Cruz, Argentina. El Calafate is a popular tourist destination, being the access point to the Perito Moreno glacier.
West towards Parque Nacional de los Glaciares (Perito Moreno)
You can just walk west along the main road (Ave. San Martín), and eventually you'll get out of town, and there's even a speedbump to slow traffic for you. If you want to get a little further away, there's a free shuttle to the Glaciarum that leaves from San Martín and 1° de Mayo every hour on the hour. From Glaciarum, you can walk to the road, and traffic will be going quickly, but everyone who passes is going to the park.
East towards everywhere else (El Chaltén, Rio Gallegos, Puerto Natales)
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 gendarmeria where traffic slows down, and it's a great place for hitching (01/2018 - gendarmeria is already closed but still it's the best point). 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. Unfortunately Argentinian and Chilean hitchhikers do not follow the rule - "came later, stand later", and often stand before you, some of them was lucky to take a car on rotunda before gendarmeria so you can try it.