In this paper, we solve numerically in the complex plane all the differential equations involved in Hartle's perturbative scheme for rapidly rotating neutron stars. We first find that some significant quantities (as the unperturbed radius and mass) remain almost invariable with respect to several trial imaginary initial values (of central density and pressure) varying over an orders-of-magnitude range. We then compare our numerical results with corresponding results of several nonperturbative methods, emphasizing on quantities describing the geometry of rapidly rotating models; we find appreciable improvement of our results in comparison with those given by the "classical" Hartle's scheme.