This paper aims at the development of a regression-aided analytical framework for modeling and analyzing the compressibility behavior of over-consolidated soils. A three-parameter rectangular hyperbola function (3P-RH) was proposed for describing the void ratio–effective stress relationship. Validation of the 3P-RH was carried out by a compiled database gathered from the literature. Simple analytical solutions were then proposed for determining the compressibility curve variables including the compression (Cc) and recompression (Cr) indices and the preconsolidation pressure (Pc), which are intended to replace the current subjective graphical method by providing consistent results. Equations for the preconsolidation pressure were derived in accordance with four common graphical constructions covering various levels of geometrical complexity (slightly to highly subjective). A probabilistic comparison among the graphical constructions was then carried out. Furthermore, a sensitivity analysis with respect to the proposed preconsolidation pressure functions was considered to evaluate the influence of the 3P-RH fitting parameters (α and β) on the preconsolidation pressure value. The proposed 3P-RH compressibility model accompanied by the suggested analytical solutions for solving the compressibility curve variables construct a unique framework for modeling the compressibility behavior of soils with an acceptable degree of accuracy and, more importantly, by a simple objective approach.