The solution of large sparse linear systems is a critical operation for many numerical simulations. To cope with the hierarchical design of modern supercomputers, hybrid solvers based on Domain Decomposition Methods (DDM) have been been proposed. Among them, approaches consisting of solving the problem on the interior of the domains with a sparse direct method and the problem on their interface with a preconditioned iterative method applied to the related Schur Complement have shown an attractive potential as they can combine the robustness of direct methods and the low memory footprint of iterative methods. In this report, we consider an additive Schwarz preconditioner for the Schur Complement, which represents a scalable candidate but whose numerical robustness may decrease when the number of domains becomes too large. We thus propose a two-level MPI/thread parallel approach to control the number of domains and hence the numerical behaviour. We illustrate our discussion with large-scale matrices arising from real-life applications and processed on both a modern cluster and a supercomputer. We show that the resulting method can process matrices such as tdr455k for which we previously either ran out of memory on few nodes or failed to converge on a larger number of nodes. Matrices such as Nachos_4M that could not be correctly processed in the past can now be efficiently processed up to a very large number of CPU cores (24,576 cores). The corresponding code has been incorporated into the MaPHyS package.