The motorway runs through the city, so it's easy to hitch northwards or southwards.
North towards Salerno
Take a circulare (local bus) to Rende, the northern part of town. From the town hall, follow via Leonardo da Vinci and then turn left into via Marconi until you get to the roundabout of the motorway exit Cosenza Nord. There are often Carabinieri controlling traffic on that roundabout, therefore a better option might be the service station Cosenza Est. Again, from Rende find via Rossini and then turn into via Londra. Shortly before arriving at the motorway, turn right, walk around some houses through some little streets; you should be able to see the petrol station through some pine trees. You can directly walk onto the petrol station, as the fences are pretty much demolished.
South towards Reggio Calabria
Southwards, find the motorway exit Cosenza Sud, it is not rare to find other hitchhikers there.
Accommodation and Sleep
There are some occupied spaces and social centres in the community of Rende. Ask your way around.