The original motivation for scaled particle theory (SPT) is to derive a simple equation of state for a bulk hard sphere (HS) fluid. It is now widely recognized that SPT provides also the surface tension of a HS fluid near a spherical hard wall, including three contributions, i.e., the planar surface tension and two bending rigidities due to the surface curvatures (integrated mean and Gauss curvatures). This conforms to the basic assumption of morphological thermodynamics. The existence of non-Hadwiger terms, i.e., higher-order curvature terms, has been evidenced recently. Augmenting SPT by only one non-Hadwiger term, i.e., third-order curvature term, allows us to obtain two new analytical theories, which not only describe very accurately bulk thermodynamic properties but also improve systematically surface ones with respect to SPT.