Walk the 7 km from the city center northward to the main road, or take the city bus (bus urbano) for $0.30 out from Avenida Napo. Ask if the bus will take a left after the crossing. If it does, stick around until the bus is completely empty and the driver wants you to get out. You'll get out on a shoulderless road, but there should be a muro (speed bump) not too far away. Use the speed bump to your advantage and catch a ride to wherever. There's a fork in the road at a town called Narupa, so make sure you get out here if your driver is going one way and you the other. Before that, there's a police control to see if you're not smuggling any tropical birds or monkeys. Mind of a Hitchhiker caught her first Ecuadorian hitch from this speed bump within 2 minutes all the way to Tena, though the drivers would drive all the way to Ambato.
The boats to Peru on the Napo River
To cross the border from Ecuador to Perú in this region, you have to take a few boats as there are no roads. Prices are as of August 2019. Hitchwiki user Mind of a Hitchhiker wrote a detailed account of this boat trip on her blog. It is possible to hitchhike the boats, but not always for free. For example, you can go with the cargo boats, as they also take passengers, but they will maybe charge you (cheaper than the actual boats).
1. From El Coca to Nuevo Rocafuerte. Leaves daily. Costs $18.75. In Nuevo Rocafuerte you can stamp out of Ecuador at the immigration office in the police station (Careful!! On August 2019, it was not possible to stamp in Rocafuerte anymore) Francois succeed to hitchhike a cargo boat there in exchange of helping them working to load and unload the Canoa.
2. From Nuevo Rocafuerte to Cabo Pantoja in Perú. Leaves during daylight hours if you pay a local. No official boat. Costs about $20 or 60 Peruvian Soles, depending on how many people, the exchange rate, and your negotiation skills. Stamp into Perú in Pantoja at the immigration office next to the police station and a huge military base. This portion is definitely hitchhikeable. Just wait at the dock and ask everybody that passes (cargo boat, privata Canoa, fast boat). Francois succeed to hitchhike a Ambulance boat in this section.
3. From Cabo Pantoja to Mazan via Santa Clotilde. This is where you get to have options: A) take the slow boat that leaves twice per month (100sols, also goes to Iquitos directly), or B) take the fast boat that leaves twice per week and includes accommodation in Santa Clotilde and some food and drinks for 200 Peruvian Soles (about $60). If you don't carry any Peruvian Soles, try to find a gringo to exchange with in Pantoja. Exchanging Peruvian Soles in Ecuador is really, really tough. No idea if it is possible to hitchhike it.
4. From Mazan to Iquitos. There's a fast boat (15 Peruvian Soles) and a slow boat (8 Peruvian Soles) and they leave every day until about 15:00. You have to walk or take a mototaxi (3 Peruvian Soles) across the peninsula of Mazan to get to the other harbor. In Iquitos you're in a big city again and there's ATMs if you need them.