Contributions to the fifth-order nonlinear optical susceptibility ${\ensuremath{\chi}}^{(5)}$ of a collection of homogeneously broadened two-level atoms that scale as ${N}^{2}{({\ensuremath{\gamma}}_{\mathrm{at}}^{(3)})}^{2}$ and ${N}^{2}{\ensuremath{\mid}{\ensuremath{\gamma}}_{\mathrm{at}}^{(3)}\ensuremath{\mid}}^{2}$, where ${\ensuremath{\gamma}}_{\mathrm{at}}^{(3)}$ is the lower-order atomic hyperpolarizability and
N is the atomic number density, are predicted theoretically. These ``cascaded'' contributions are a consequence of local-field effects. We determine them from a fifth-order solution of the Lorentz-Maxwell-Bloch equations. They are missing from a straightforward generalization of Bloembergen's result for the local field correction to the second order nonlinearity, but are recovered by a careful application of his general approach. We find that at high atomic densities $(N>{10}^{15}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3})$ the value of the cascaded third-order contribution can be as large as the ``direct'' fifth-order term in the expression for the fifth-order susceptibility.