The coupled reference interaction site model -hyper-netted chain (RISM-HNC)/ simulation methodology determines solvation free energies as a function of the set of all radial distribution functions of solvent atoms about atomic solute sites. These functions are determined from molecular dynamics (MD) or Monte Carlo (MC) simulations rather than from solving the RISM and HNC equations iteratively. Previous applications of the method showed that it can predict relative free energies of solvation for small solutes accurately. However, the errors scale with the system size. In this study, we propose the use of the hard-sphere free energy as the reference and a linear response approximation to improve the performance, i.e., accuracy and robustness, of the method, particularly removing the size dependency of the error. The details of the new formalism are presented. To validate the proposed formalism, solvation free energies of N-methylacetamide and methylamine are computed using the new RISM-HNC-based expressions in addition to a linear response expression, which are compared to previous thermodynamic integration and thermodynamic perturbation results performed with the same force field. Additionally, free energies of solvation for cyclohexane, pyridine, benzene and derivatives, and other small organic molecules are calculated and compared to experimental values.