Résumé : In this paper, we present an hybrid solver for linear systems that combines a Krylov subspace method as accelerator with some overlapping domain decomposition method as preconditioner. The preconditioner uses an explicit formulation associated to one iteration of the classical multiplicative Schwarz method. To avoid communications and synchronizations between subdomains, the Newton-basis GMRES implementation is used as accelerator. This requires to divide the computation of the orthonormal basis in two steps: the preconditioned Newton basis is computed then it is orthogonalized. The first step is merely a sequence of matrix-vectors and solution of linear systems associated to subdomains. We describe the fine-grained parallelism that is used in these kernel operations. The second step uses a parallel implementation of dense $QR$ factorization on the resulted basis. At each application of the preconditioner operator, local systems associated to the subdomains are solved with some accuracy depending on the global physical problem. We show that this step can be further parallelized with calls to external third-party solvers. To this end, we define two levels of parallelism in the solver: the first level is intended for the computation and the communication across all the subdomains; the second level of parallelism is used inside each subdomain to solve the smaller linear systems induced by the preconditioner. Numerical experiments are performed on several problems to demonstrate the benefits of such approaches, mainly in terms of global efficiency and numerical robustness.