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The relaxed physical factorization (RPF) preconditioner is a recent algorithm allowing for the efficient and robust solution to the block linear systems arising from the three-field displacement-velocity-pressure formulation of coupled poromechanics. For its application, however, it is necessary to invert blocks with the algebraic form $\hat{C} = ( C + βF F^T)$, where $C$ is a symmetric positive definite matrix, $FF^T$ a rank-deficient term, and $β$ a real non-negative coefficient. The inversion of $\hat{C}$, performed in an inexact way, can become unstable for large values of $β$, as it usually occurs at some stages of a full poromechanical simulation. In this work, we propose a family of algebraic techniques to stabilize the inexact solve with $\hat{C}$. This strategy can prove useful in other problems as well where such an issue might arise, such as augmented Lagrangian preconditioning techniques for Navier-Stokes or incompressible elasticity. First, we introduce an iterative scheme obtained by a natural splitting of matrix $\hat{C}$. Second, we develop a technique based on the use of a proper projection operator annihilating the near-kernel modes of $\hat{C}$. Both approaches give rise to a novel class of preconditioners denoted as Enhanced RPF (ERPF). Effectiveness and robustness of the proposed algorithms are demonstrated in both theoretical benchmarks and real-world large-size applications, outperforming the native RPF preconditioner.