North towards Las Flores, Rodeo or San Juan city
If you're coming from Calingasta town, head east towards the bridge, cross it and go to the Bandera Blanca gas station you see. Try to be there as early as possible, as many cars don't hit the road during the heat of the day. There's a nice shadow in the morning hours during summer. You can stand between the gas station and road with your stuff and walk toward the gas station to ask people, or put up your thumb when they're just passing by. Mind of a Hitchhiker was there at 7:00 in the morning in January 2017 and got picked up really quickly. If someone is going to San Juan and you're going to Rodeo or vice versa, get out at the roundabout where the RN 149 meets the RP 436. Every local knows it. There's a little chapel there, but no shadow, so put on some clothes till someone picks you up from there. If you're going to Rodeo, it might be best to ask for Iglesia (the name of the department) or Las Flores, as those are right before Rodeo. It's smooth sailing from those towns onward.
Las Flores is where the Ruta Nacional 150 leads into Chile, at the Paso Agua Negra, 4.780 meters above sea level. The Argentinian immigration and customs is 3 km from the nearest YPF gas station in Las Flores, west of the crossing towards Rodeo.
South towards Barreal
Follow the previously mentioned advice, but ask at the Bandera Blanca gas station for rides going to Barreal. You will miss out on traffic here though, so you can also position yourself at the crossing where the traffic goes to Barreal. The majority of traffic on this stretch of the RN 149 is working people who are commuting between these two villages, so the wait can never be that long.
Accommodation and Sleep
There's a lot of Pampa landscape between Calingasta and Barreal. Pitch your tent in a secluded spot far from roads and villages and you'll have a decent night of sleep under the dark skies.