  Accurate waveform models are crucial for gravitational-wave data analysis,
and since spin has a significant effect on the binary dynamics, it is important
to improve the spin description in these models. In this paper, we derive the
spin-orbit (SO) coupling at the fifth-and-a-half post-Newtonian (5.5PN) order.
The method we use splits the conservative dynamics into local and
nonlocal-in-time parts, then relates the local-in-time part to gravitational
self-force results by exploiting the simple mass-ratio dependence of the
post-Minkowskian expansion of the scattering angle. We calculate the nonlocal
contribution to the 5.5PN SO dynamics to eighth order in the small-eccentricity
expansion for bound orbits, and to leading order in the large-eccentricity
expansion for unbound orbits. For the local contribution, we obtain all the
5.5PN SO coefficients from first-order self-force results for the redshift and
spin-precession invariants, except for one unknown that could be fixed in the
future by second-order self-force results. However, by incorporating our 5.5PN
results in the effective-one-body formalism and comparing its binding energy to
numerical relativity, we find that the remaining unknown has a small effect on
the SO dynamics, demonstrating an improvement in accuracy at that order.
