Esquel is a city in Chubut, Argentina located along the famous Ruta 40. When traveling south through Patagonia, many people recommend heading east at Esquel to continue south along the Ruta 3, which is paved and has much more traffic. Hitchhiking is common in this area, and waits tend to be short.
Leaving town, there's just one road north, Av. Ameghino, just walk to the roundabout past the army base, and you can try there. If you encounter too much local traffic, or if you prefer talking with people, about 1 km further down is a petrol station where you can try your luck. If you're up for a good walk or if you can get a short ride, a few kilometers down the road is a police checkpoint, which is a great place to get a ride. If you're trying to get to Trelew and find too much traffic at the checkpoint is going to El Bolsón, about one or two more kilometers down the road is where it splits in one direction to El Bolsón and the other to Trelew. Cars are going kind of fast there, but they'll all be going where you need to go.
South towards Trevelin/Parque Nacional de los Alerces
Walk south along Av. Alvear (the main avenue) all the way to the end (about 2 km from the center), turn right and walk for about five more blocks to a roundabout. There's a bus stop there, and it's a perfect spot with lots of hitchhiking going on. If you're going to Los Alerces, there's a junction where the road splits about ten or fifteen kilometers away. People going to Trevelin will offer to take you there, but it might not be worth it: traffic coming from Esquel doesn't slow down there, and there are other routes to the park from Trevelin so there's not much traffic passing from that direction. Once in the park, hitching is really easy if you're alone during high season.