We revisit the successful scaled particle theory (SPT) of hard particle fluids, originally developed by Reiss, Frisch, and Lebowitz (J. Chem. Phys. 1959, 31, 369). In the initial formulation of SPT, five exact conditions were derived that constrained the form of the central function G. Only three of these conditions, however, were employed to generate an equation of state. Later, the number of relations used to determine G was increased to five (Mandell, M. J.; Reiss, H., J. Stat. Phys. 1975, 13, 113). The resulting equation of state was an improvement over the original formulation, although its accuracy was still limited at high densities. In an effort to increase the accuracy of SPT predictions, we propose two new formally exact conditions on the form of G. These sixth and seventh conditions relate exactly known derivatives of G to the slope and curvature of the hard sphere radial distribution function at contact, g'(sigma(+)) and g"(sigma(+)), respectively. To apply the new conditions, we derive, again within the framework of SPT, physically and geometrically based approximations to g'(sigma(+)) and g"(sigma(+)). These additional restrictions on the function G yield markedly improved predictions of the pressure, excess chemical potential, and work of cavity formation for the hard sphere fluid, now making SPT competitive with other existing equations of state.