  Future gravitational-wave detectors, especially the Laser Interferometer
Space Antenna (LISA), will be sensitive to black hole binaries formed in
astrophysical environments that promote large eccentricities and spin
precession. Gravitational-wave templates that include both effects have only
recently begun to be developed. The Efficient Fully Precessing Eccentric (EFPE)
family is one such model, covering the inspiral stage with
small-eccentricity-expanded gravitational-wave amplitudes accurate for
eccentricities $e < 0.3$. In this work, we extend this model to cover a larger
range of eccentricities. The new EFPE_ME model is able to accurately represent
the leading-order gravitational-wave amplitudes to $e \leq 0.8$. Comparing the
EFPE and the EFPE_ME models in the LISA band, however, reveals that there is no
significant difference when $e_0 \leq 0.5$ for binaries at 4 years before
merger, as radiation reaction circularizes supermassive black hole binaries too
quickly. This suggests that the EFPE model may have a larger regime of validity
in eccentricity space than previously thought, making it suitable for some
inspiral parameter estimation with LISA data. On the other hand, for systems
with $e_0 > 0.5$, the deviations between the models are significant,
particularly for binaries with total masses below $10^5\, \mathrm{M}_{\odot}$.
This suggests that the EFPE_ME model will be crucial to avoid systematic bias
in parameter estimation with LISA in the future, once this model has been
hybridized to include the merger and ringdown.
