Hard spherocylinders (cylinders of length L and diameter D capped at both ends with two hemispheres) provide a suitable model for investigating entropy-driven mesophase formations in real colloidal fluids that are composed of rigid rodlike molecules. We performed extensive Monte Carlo simulations of this model fluid for elongations in the range of 3 less than or equal to 5 L/D less than or equal to 5 and up to L/D = 20 to investigate the relative importance of translational and orientational correlations allowing for the emergence of nematic or smectic order in the framework of the so-called residual multiparticle entropy (RMPE). The vanishing of this quantity, which includes the re-summed contributions of all spatial correlations involving more than two particles, signals the, structural changes that take place at increasing densities in the isotropic fluid. We found that the ordering thresholds detected through the zero-RMPE condition systematically correlate with the corresponding phase-transition points, whatever the nature of the higher-density phase coexisting with the isotropic fluid. PACS 64.60.-i, 64.70.Md, 61.20.Ja, 65.40.Gr.