We present new results for the hard particle system (2D and 3D cases) in the low density branch of the equation of state that provide substantial improvement over results given by us in an earlier work. The well known low density limit of the equation of state allows an accurate determination of a parameter m(eta), a function of the packing fraction eta, which then allows an accurate computation of the equation of state at higher densities throughout the low density branch. Our approach therefore provides an extrapolation scheme in which the known behavior of the hard particle fluid in some density regime provides a "signature" via the parameter m(eta) from which the fluid behavior at other densities is predictable. We note that the developments in this paper apply as well to arbitrary equilibrium systems provided "m" may be appropriately chosen as a function of density and temperature. Also, unlike most other extrapolation schemes, our approach is of a systematic nature, not involving ad hoc approximations or assertions that are not rigorously founded. Extensions of our approach to the high density branch of the equation of state, as well as computations of error bounds for our results are also discussed.