The region with the most public access and conentration of different waterways close to tourist amenities would be the Junin/San Martin region in Argentina. It's relatively easy to rent a car, hire a taxi, or even hitch hike to the river from either town. Junin is less touristy than San Martin and closer to the Chimehuin river, but isn't as attractive of a town.
There are of course other regions to fish, but from a DIY perspective it's pretty hard to beat this region of Patagonia for convenience.
Hope that helps!