  The SecularMultiple code, presented in two previous papers of this series,
integrates the long-term dynamical evolution of multiple systems with any
number of bodies and hierarchical structure, provided that the system is
composed of nested binaries. In the formalism underlying SecularMultiple, we
previously averaged over all orbits in the system. This approximation
significantly speeds up numerical integration of the equations of motion,
making large population synthesis studies possible. However, the orbit
averaging approximation can break down when the secular evolution timescale of
the system is comparable to or shorter than any of the orbital periods in the
system. Here, we present an update to SecularMultiple in which we incorporate
hybrid integration techniques, and orbit-averaging corrections. With this
update, the user can specify which orbits should be integrated directly
(without averaging), or assuming averaged orbits. For orbits that are
integrated directly, we implemented two integration techniques, one which is
based on the regularised Kustaanheimo-Stiefel equations of motion in element
form. We also implemented analytical orbit-averaging corrections for pairwise
interactions to quadrupole order. The updates presented here provide more
flexibility for integrating the long-term dynamical evolution of hierarchical
multiple systems. By effectively combining direct integration and orbit
averaging the long-term evolution can be accurately computed, but with
significantly lower computational cost compared to existing direct N-body
codes. We give a number of examples in which the new features are beneficial.
Our updated code, which is written in C++ supplemented with a user-friendly
interface in Python, is freely available.
