For Chile, there is a border crossing to Chile Chico. At the time of research, public transportation across the border was suspended. Inquire locally, as this situation could likely change. It's 1.5km from Los Antiguos to the Argentine border, then 1km further to the Chilean border, with Chile Chico 5km beyond.
A Chilean ferry run by Somarco crosses Lago General Carrera daily from Chile Chico to Puerto Ingeniero Ibañez almost daily, a big shortcut to Coyhaique.