Ecuador is a country in South America.
The bomberos (firemen) are more hospitable in Ecuador than in other countries. Drop by the firehouse and ask if they have a space for you to lay your blankets and pitch your tent. Usually they will let you use their bathrooms and kitchen too.
As a last resort, there are police in bus terminals all night, if you don't mind sleeping under the florescent lights.
Meet fellow travellers on hospitality exchange networks: Trustroots
In addition to the internet cafes, you may also find Infocentros in small towns around the country. These are government-funded facilities with free computers with internet access. You just need to show your ID to use a computer (a foreign driver's license worked fine). Look out for signs that say "Infocentro <Town name>" along the highway. There is also a map of Infocentros on their website (in spanish).
Ecuador only borders two countries: Peru to the South and East and Colombia to the North and East. In the Amazon region of Ecuador, there's some border crossings by river boat that aren't really good for hitchhiking, but might provide a great challenge.
Ecuador has one of the most chill visa policies in the world, therefore many people with "difficult" passports can also come and enjoy the country for 90 days.
All border crossings are ordered roughly from West to East. Ecuador and Perú had a border dispute as recent as 1999, so perhaps be sensitive on the issue. This is not a comprehensive list of border crossings, as there are many more, but most of these will have immigration offices to process international visitors.
Huaquillas to Aguas Verdes is probably a local border crossing between the two countries, not suitable for foreign passports. The border is the bridge over the Zarumilla river. The location of immigration offices is unknown. Please add information if you've hitched this. Perhaps take the next border crossing instead:
Puente Internacional Zarumilla is a major border crossing between roughly Ciudad Astral in Ecuador and Zarumilla in Perú. The location of the Ecuadorian customs and immigration is about 2 km from the actual border. The location of the border control on the Peruvian side is unknown. Please add info if you've hitched this!
El Alamor between Zapotillo in Ecuador on the E25 and Lancones in Perú on the 1NN. The immigration offices are about 600 meters apart across a small bridge over the Rio Alamor. Please add info if you hitchhiked this!
Puente Internacional Macará from Macará in Ecuador on the E35 to Suyo in Perú on the 1NL. It seems to be the "real" Panamerican Highway border crossing between the two countries. The immigration offices are about 500 meters apart. Please add info if you hitched this!
Rio Canchis/La Balza from Zumba, Ecuador on the E682, to Namballe, Perú on the 5N. Immigration offices are 400 meters apart over a bridge. Let us know if you hitched this border!
Río Santiago perhaps there is a river boat from Jempekat/Soldado Monge in Ecuador via the Rio Santiago and the Rio Marañon to Saramiriza in Perú. There seems to be both a Peruvian and an Ecuadorian settlement at the border, which is perhaps a military base or an immigration office. Let us know if you crossed the border here!
Río Pastaza perhaps there is a river boat from between Puyo and Macas in Ecuador down the Río Pastaza (border town named Andoas), to San Lorenzo/Yurimaguas in Perú. Let us know if you're aware this route exists!
Río Napo between Nuevo Rocafuerte/El Coca in Ecuador and Cabo Pantoja/Iquitos in Perú. Mind of a Hitchhiker crossed this border in May 2017 in opposite direction and wrote extensively about this trip and made a video here! Immigration offices are in Nuevo Rocafuerte and Cabo Pantoja. IMPORTANT: when trying to leave Ecuador to Colombia via Tulcán border crossing, it turned out Mind of a Hitchhiker's entry stamp from Nuevo Rocafuerte was not entered in the electronic system - guess what, the connection in the jungle isn't that great! She was turned away to make copies of her passport (three of the main page, three of the entry stamp page) and return. The lady who copies your passport is in a little booth 40 meters from the immigration office. The six pages will cost you $0.60 and then the officer is happy. Realize when crossing the Nuevo Rocafuerte border, you'll need to take a little more time to exit Ecuador because of your silly (but great) stamp!
Güeppi is the (northernmost) Peruvian town on the tri-border with Ecuador and Colombia on the banks of the river Putumayo. There is a Peruvian military base here. To enter via Ecuador, enter the Putumayo river from the town Puerto El Carmen de Putumayo, or another village at the river connected by road. Getting here via any of the Ecuadorian or Colombian rivers would be both an incredible adventure and a hassle. One can follow the Río Putumayo downstream until Tarapaca in Colombia on the border with Brazil. Eventually, the river will enter the majestic Amazon river. Let us know if you did it!
All border crossings are ordered roughly from West to East. Ecuador was formed from the former country Gran Colombia and the flags of Ecuador, Colombia and Venezuela are nearly the same design, indicating their common history. The following list of border crossings isn't comprehensive.
Mataje roughtly from San Lorenzo in Ecuador on the E10 to Espriella/Tumaco in Colombia on the ruta 10. WARNING! This border crossing is rather new, and at the time of research (June 2017) it wasn't clear yet if the border crossing had been opened yet. It was announced in 2014 and the project would take two years to finish, so ideally it should be done already. It crosses the Río Mataje. Let us know if this fresh border crossing has been opened to international travelers already and how it was to hitchhike it!
Tufiño - Chiles from Tufiño in Ecuador on the E182, to Chiles in Colombia. There is a bridge across the Carchi river, which forms the border. It is unknown if and where the immigration offices are located and if it's possible to cross on a foreign passport. Let us know if you hitched this border!
Puente Internacional Rumichaca between Tulcán in Ecuador and Ipiales in Colombia. This is the "official" Panamerican Highway border crossing between the two countries, and the one most certainly accepting foreign passports. The distance between immigration offices is 400 meters. There's a bridge crossing the Carchi river. Please add information if you hitched this!
Urbina a few kilometers from the Rumichaca border crossing is the smaller border crossing in the Ecuadorian town of Urbina at the river of the same name. The nearest town on the Colombian side is again Ipiales. It is unknown where the Colombian stamp-in immigration office is at. Let us know if you hitched this!
Puente Internacional San Miguel between Nueva Loja (also known as Lago Agrio) in Ecuador on the E45, and La Hormiga in Colombia on the Ruta 45. The immigration offices are at the San Miguel river bridge and are 200 meters apart. Please add information if you hitched this border!
Puerto Asís isn't a real border, but it is a town located at the Putumayo river that forms part of the border with Ecuador. It is likely possible that there are river boats from here to Puerto El Carmen de Putumayo in Ecuador, the tri-border with Perú called Güeppi, and far away places in Colombia like Tarapaca (near Leticia). Eventually, the Putumayo river will become part of the Amazon river in Brazil and end in the Atlantic Ocean, if you're looking at the big picture. Please add any information you have if you have taken a river boat border crossing in this area!
Centros de Salud, found in most cities and towns, give out free condoms, and probably healthcare too.