  We analytically compute, through the six-and-a-half post-Newtonian order, the
second-order-in-eccentricity piece of the Detweiler-Barack-Sago gauge-invariant
redshift function for a small mass in eccentric orbit around a Schwarzschild
black hole. Using the first law of mechanics for eccentric orbits [A. Le Tiec,
Phys. Rev. D {\bf 92}, 084021 (2015)] we transcribe our result into a
correspondingly accurate knowledge of the second radial potential of the
effective-one-body formalism [A. Buonanno and T. Damour, Phys. Rev. D {\bf 59},
084006 (1999)]. We compare our newly acquired analytical information to several
different numerical self-force data and find good agreement, within estimated
error bars. We also obtain, for the first time, independent analytical checks
of the recently derived, comparable-mass fourth-post-Newtonian order dynamics
[T. Damour, P. Jaranowski and G. Shaefer, Phys. Rev. D {\bf 89}, 064058
(2014)].
