  The (first-order) gravitational self-force correction to the spin-orbit
precession of a spinning compact body along a slightly eccentric orbit around a
Schwarzschild black hole is computed through the ninth post-Newtonian order,
improving recent results by Kavanagh et al. [Phys.\ Rev.\ D {\bf 96}, 064012
(2017).] This information is then converted into its corresponding
Effective-One-Body counterpart, thereby determining several new post-Newtonian
terms in the gyrogravitomagnetic ratio $g_{S*}$.
