We modify the site-site as well as three-dimensional (3D) versions of the reference interaction site model (RISM) integral equations with the hypernetted chain (HNC) closures by adding a repulsive bridge correction (RBC). The RBC treats the overestimation of water ordering around a hydrophobic solute in the RISM/HNC approximation, and thus refines the entropic component in the hydration free energy. We build up the bridge functions on r(-12) repulsive core potentials, and propose RBC expressions for both the site-site and 3D-RISM approaches. To provide fast calculation, we obtain the excess chemical potential of hydration by using the thermodynamic perturbation theory (TPT). The site-site RISM/HNC+RBC as well as 3D-RISM/HNC+RBC approaches are applied to calculate the structure and thermodynamics of hydration of rare gases and alkanes in ambient water. For both approaches, the RBC drastically improves the agreement of the hydration chemical potential with simulation data and provides its correct dependence on the solute size. For solutes of a nonspherical form, the 3D treatment yields the hydration structure in detail and better fits simulation results, whereas the site-site approach is essentially faster. The TPT approximation gives the hydration thermodynamics in good qualitative agreement with the exact results of the thermodynamic integration, and substantially reduces computational burden. The RBC-TPT approximation can improve the predictive capability of the hybrid algorithm of a generalized-ensemble Monte Carlo simulation combined with the site-site RISM theory, used to describe protein folding with due account for the water effect at the microscopic level. The RBC can be optimized for better fit to reference simulation data, and can be generalized for solute molecules with charged groups.