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Rosario is Argentina's third biggest city after Buenos Aires and Cordoba. It is part of Santa Fe province and is located inland at the Paraná river, which sees a lot of shipping traffic. Only the west side of the river has been built on, while the east side is largely filled with swampy islands with semi-inhabited buildings and loads of nature (and mosquitoes).
South East towards Buenos Aires
A good place to hitchhike from Rosario to Buenos Aires is the peaje en arroyo seco. You can go there by a bus from the bus station, that travels up to San Nicolas de los Arroyos. You'll have to talk to the bus driver and explain to him that you want to get dropped at the "peaje".
West towards Cordoba
One can take a city bus towards the entrance of the Autopista Ernesto Guevara (road number Autopista 9) on the on-ramp that intersects Avenida Provincias Unidas. There's a traffic light.
Mind of a Hitchhiker took a city bus towards Villa Fisherton and then started hitchhiking the parallel Ruta 9 (not a highway. Also known as the 1V09) via the town of Funes towards the final town of Roldán. It's a hassle, but from Roldán one can enter the Petrobras gas station via the back entrance and that gas station is on the Autopista 9. Loads of effort, but the first attempt at searching a ride yielded a direct hitch to Cordoba city. This Petrobras is also open 24/7 and has a shop, bathrooms with toilet paper and comfy couches.
East towards Victoria, Gualeguay, Gualeguaychu in Entre Rios province and Uruguay
A giant bridge crosses the giant Paraná river on what is officially called the Ruta 174, Conexión Vial Rosario - Victoria. The best way to enter might be by taking the bus to the Makro (a bulk-buy shop) on the outskirts of town and getting out before crossing the giant highway. There's two on-ramps within 1km of each other from the road called Juan Pablo II. The traffic that enters here has only one chance to leave the highway before the bridge from here, so it's your best bet. Write a sign for Victoria and wait till you catch something! The bridge, named Puente Nuestra Señora del Rosario is a toll-bridge, but the toll is only paid on the east end of the bridge and not the west (Rosario) side. There's a little parking, so one can even decide to switch rides here to find something that will go past Victoria.
Many people will tell you Rosario is the most dangerous city of Argentina. Make up your own mind about the city.
Mind of a Hitchhiker went freecamping on the Isla Verde, on the west side of the Paraná River and 14 kilometers north of Rosario's centre. Take the bus 153 Negro and get out at wherever the bus driver says is the pier with the boat towards Isla Verde (La muelle con la lancha a Isla Verde). The boats run every day from 8:00 til 19:00 and a return ticket costs 60 pesos. One can also try to hitchhike a boat or a kayak from one of the many beaches and ports along this route, but that really depends on your sociability and how much stuff you carry (and if it may get wet). On the island one can camp, depending on where you arrive. The land is owned by people, but upon stating a foreign nationality, you might be invited to stay for free as a guest. There's barbecues (parillas) everywhere and many, many (farm) animals walking about. There's a bathroom and a shop that sells anti-mosquito cream (wise choice), bottled water and some food things, like fresh eggs from the wandering chickens. It's better not to go here on the weekend, as the entire city crosses in their kayaks on the sunny days.
Ernesto "Che" Guevara was born in this city. There's a city park named after him near the railroads where one could try to camp at night. Might be a little sketchy.