As the computational power of high performance computing (HPC) systems continues to increase by using a huge number of CPU cores or specialized processing units, extreme-scale applications are increasingly prone to faults. As a consequence, the HPC community has proposed many contributions to design resilient HPC applications, may these contributions be system-oriented, theoretical or numerical. In this study we consider an actual fully-featured parallel sparse hybrid (direct/iterative) linear solver, \maphys, and we propose numerical remedies to design a resilient version of the solver. The solver being hybrid, we focus in this study on the iterative solution step, which is often the dominant step in practice. We furthermore assume that a separate mechanism ensures fault detection and that a system layer provides support for setting back the environment (processes, \ldots) in a running state. The present manuscript therefore focuses on (and only on) strategies for recovering lost data \emph{after} the fault has been detected (a separate concern out of the scope of this study) and \emph{once} the system is back in a running state (another separate concern not studied here either). The numerical remedies we propose are twofold. Whenever possible, we exploit the natural data redundancy between processes from the solver to perform exact recovery through clever copies over processes. Otherwise, data that has been lost and is not available anymore on any process is recovered through a so-called interpolation-restart mechanism. This mechanism is derived from~\cite{aggr:13} to carefully take into account the properties of the target hybrid solver. These numerical remedies have been implemented in the \maphys parallel solver so that we can assess their efficiency on a large number of processing units (up to $12,288$ CPU cores) for solving large-scale real-life problems.