Domain-decomposition (DD) methods are used in most, if not all, modern parallel implementations of finite element modeling software. In the solver stage, the algebraic additive Schwarz (AAS) domain-decomposition preconditioner represents a fundamental component and its performance and scalability are key to the overall performance of the solution process. The established approach to construct the preconditioner in a parallel MPI setting is with a 1-to-1 correspondence between the number of MPI processes and the number of AAS subdomains. In this paper, we describe our attempts to extend this paradigm with the possibility to assign more than one MPI process per AAS subdomain, with the goal of improving the overall performance of the AAS preconditioner on supercomputers with multicore nodes. We discuss the implementation of the new AAS preconditioner framework, based on two levels of MPI parallelism, and the performance of different subdomain solver strategies. Finally, we examine the behavior of our novel approach for a series of benchmark problems, performed with the LifeV parallel finite element library.