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We calculate the eccentricity dependence of the high-order post-Newtonian(PN) expansion of the spin-precession invariant $\psi$ for eccentric-orbitextreme-mass-ratio inspirals with a Schwarzschild primary. The series iscalculated in first-order black hole perturbation theory through directanalytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using acode written in \textsc{Mathematica}. Modes with small values of $l$ are foundvia the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism forsolutions to the Regge-Wheeler equation. Large-$l$ solutions are found byapplying a PN expansion ansatz to the Regge-Wheeler equation. Previous work hasgiven $\psi$ to 9.5PN order and to order $e^2$ (i.e., the near circular orbitlimit). We calculate the expansion to 9PN but to $e^{16}$ in eccentricity. Itproves possible to find a few terms that have closed-form expressions, all ofwhich are associated with logarithmic terms in the PN expansion. We alsocompare the numerical evaluation of our PN expansion to prior numericalcalculations of $\psi$ in close orbits to assess its radius of convergence. Wefind that the series is not as rapidly convergent as the one for the redshiftinvariant at $r \simeq 10M$ but still yielding $\sim 1\%$ accuracy foreccentricities $e \lesssim 0.25$.

Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals to 9PN and $e^{16}$