  We calculate the eccentricity dependence of the high-order post-Newtonian
(PN) expansion of the spin-precession invariant $\psi$ for eccentric-orbit
extreme-mass-ratio inspirals with a Schwarzschild primary. The series is
calculated in first-order black hole perturbation theory through direct
analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a
code written in \textsc{Mathematica}. Modes with small values of $l$ are found
via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for
solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by
applying a PN expansion ansatz to the Regge-Wheeler equation. Previous work has
given $\psi$ to 9.5PN order and to order $e^2$ (i.e., the near circular orbit
limit). We calculate the expansion to 9PN but to $e^{16}$ in eccentricity. It
proves possible to find a few terms that have closed-form expressions, all of
which are associated with logarithmic terms in the PN expansion. We also
compare the numerical evaluation of our PN expansion to prior numerical
calculations of $\psi$ in close orbits to assess its radius of convergence. We
find that the series is not as rapidly convergent as the one for the redshift
invariant at $r \simeq 10M$ but still yielding $\sim 1\%$ accuracy for
eccentricities $e \lesssim 0.25$.
