If you liked Rosario you'll probably like Córdoba, which in my opinion is more beautiful. It has a nice old town (with some Unesco World Heritage buildings), shopping, and since it's a university city it also has many bars and clubs. Plus there are many small towns near the city that make good day trips.
Tucumán is not a great city. It's smaller than Córdoba and Rosario, and quite run down and chaotic. It has some nice surroundings (San Javier) but most of the nice places are further away, thus requiring more time than just a day trip.
May I suggest you consider Salta, which has a nice colonial old town and many attractions nearby. Since it's a very touristic place, lots of tour operators can arrange trips from there. Córdoba, in that regard, is not so touristic, so you'll have to rely on public transportation or rent a car.