A good reference is to investigate the Huella Andina 600km trail and all surrounding areas. Through the Andes Precordillera of Chubut, Rio Negro, and Neuquén Provinces. The is one very long set of national and provincial parks. Recently we did a peak from Villa Traful. It is a very small town on a nice lake, which had been much smaller in the past. Regardless of it's size, the summer season will bring a lot of visitors. Wikiloc.com has lots of community uploaded trails anywhere in Argentina. I have driven, camped, and hiked many of these places via Ruta 40 over a number of years if you have specific questions.

South to North:
Hiking center in Chubut is the town of Esquel. Esquel might be a good option. Los Alerces National Park
Rio Negro has El Bolson and San Carlos de Barliloche. Club Andino maintains the trails and refugios throughout a large area much greater than Bariloche. Nahuel Huapi national park
Neuquén has most popular towns of Villa La Angostura and San Martin de Los Andes. Then Villa Traful, Junin de los Andes. Lanín National Park, Alumine, etc... This 500km of lakes, mountain & towns continue until Chos Malal-Varvarco-Volcano Dumuyo.

https://huellaandinatrail.com/stages/ (english) - There is a link to a google drive PDF download which includes all the maps as a marketing brochure
Wikiloc of a Huella Andina trail in Villa Traful
Farthest north in Neuquén is Varvarco including the Dumuyo Volcano
The list of all these Patagonia national parks

There are also the big hikes of the Mendoza Cordon del Plata. This is ambitious if coming from El Calafate with only a total of 8-12 days. Also in Mendoza province is San Rafael hikes near Las Leñas ski area of Malargüe.