  Modern inference schemes for the neutron star equation of state (EoS) require
large numbers of stellar models constructed with different EoS, and these
stellar models must capture all the behavior of stable stars. I introduce
termination conditions for sequences of stellar models for cold, non-rotating
neutron stars that are guaranteed to identify all stable stellar configurations
up to arbitrarily large central pressures along with an efficient algorithm to
build accurate interpolators for macroscopic properties. I explore the behavior
of stars with both high- and low-central pressures. Interestingly, I find that
EoS with monotonically increasing sound-speed can produce multiple stable
branches (twin stars) and that large phase transitions at high densities can
produce stable branches at nearly any mass scale, including sub-solar masses,
while still supporting stars with $M > 2\,M_\odot$. I conclude with some
speculation about the astrophysical implications of this behavior.
