  We analytically compute, to linear order in the mass-ratio, the "geodetic"
spin precession frequency of a small spinning body orbiting a large
(non-spinning) body to the eight-and-a-half post-Newtonian order, thereby
extending previous analytical knowledge which was limited to the third
post-Newtonian level. These results are obtained applying analytical
gravitational self-force theory to the first-derivative level generalization of
Detweiler's gauge-invariant redshift variable. We compare our analytic results
with strong-field numerical data recently obtained by S.~R.~Dolan et al.
[Phys.\ Rev.\ D {\bf 89}, 064011 (2014)]. Our new, high-post-Newtonian-order
results capture the strong-field features exhibited by the numerical data. We
argue that the spin-precession will diverge as $\approx -0.14/(1-3y)$ as the
light-ring is approached. We transcribe our kinematical spin-precession results
into a corresponding improved analytic knowledge of one of the two
(gauge-invariant) effective gyro-gravitomagnetic ratios characterizing
spin-orbit couplings within the effective-one-body formalism. We provide
simple, accurate analytic fits both for spin-precession and the effective
gyro-gravitomagnetic ratio. The latter fit predicts that the
linear-in-mass-ratio correction to the gyro-gravitomagnetic ratio changes sign
before reaching the light-ring. This strong-field prediction might be important
for improving the analytic modeling of coalescing spinning binaries.
