  We extend our study of the extent of the regions within the $\alpha$ Centauri
AB star system where small planets are able to orbit for billion-year
timescales (Quarles & Lissauer 2016, AJ 151, 111) to investigate the effects of
minimizing the forced eccentricity of initial trajectories. We find that
initially prograde, circumstellar orbits require a piecewise quadratic function
to accurately approximate forced eccentricity as a function of semimajor axis,
but retrograde orbits can be modeled using a linear function. Circumbinary
orbits in the $\alpha$ Centauri AB system are less affected by the forced
eccentricity. {Planets on circumstellar orbits that begin with eccentricity
vectors near their forced values are generally stable, up to $\sim$$10^9$ yr,
out to a larger semimajor axis than are planets beginning on circular orbits.
The amount by which the region of stability expands is much larger for
retrograde orbits than it is for prograde orbits. The location of the stability
boundary for two planet systems on prograde, circular orbits is much more
sensitive to the initial eccentricity state than it is for analogous single
planet systems.
