HPC systems are a critical resource for scientific research. The increased demand for computational power and memory ushers in the exascale era, in which supercomputers are designed to provide enormous computing power to meet these needs. These complex supercomputers consist of numerous compute nodes and are consequently expected to experience frequent faults and crashes. Mathematical solvers, in particular, iterative linear solvers are key building block in numerous large-scale scientific applications. Consequently, supporting the recovery of distributed solvers is necessary for scaling scientific applications to exascale platforms. Previous recovery methods for iterative solvers are based on Checkpoint-Restart (CR), which incurs high fault tolerance overhead, or intrinsic fault tolerance, which require extra computation time to converge after failures. Exact state reconstruction (ESR) was proposed as an alternative mechanism to alleviate the impact of frequent failures on long-term computations. ESR has been shown to provide exact reconstruction of the computation state while avoiding the need for costly checkpointing. However, ESR currently relies on volatile memory for fault tolerance, and must therefore maintain redundancies in the RAM of multiple nodes, incurring high memory and network overheads. Recent supercomputer designs feature emerging non-volatile RAM (NVRAM) technology. This paper investigates how NVRAM can be utilized to devise an enhanced ESR-based recovery mechanism that is more efficient and provides full resilience. Our mechanism, called in-NVRAM ESR, is based on a novel MPI One-Sided Communication (OSC) over RDMA implementation, and provides full resiliency while significantly reducing both the memory footprint and the time overhead in comparison with the original ESR design (in-RAM ESR).