  The use of hyperasymptotics and the Weniger transformation has been proposed,
in a joint fashion, for decoding the divergent asymptotic series generated by
the steepest descent on a wide class of saddle-point integrals {evaluated
across Stokes sets} [R. Borghi, Phys. Rev. E {\bf 78,} 026703 (2008)]. In the
present sequel, the full development of the H-WT up to the second order in H is
derived. Numerical experiments, carried out on several classes of saddle-point
integrals, including the swallowtail diffraction catastrophe, show the
effectiveness of the 2nd-level H-WT, in particular when the integrals are
evaluated beyond the asymptotic realm.
