  We give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be
intrinsically ergodic, along with sufficient conditions for every subshift
factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that
every subshift factor of a $\beta$-shift is intrinsically ergodic, which
answers an open question included in Mike Boyle's article "Open problems in
symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe
an application of our conditions to more general coded systems. One novelty of
our approach is the introduction of a new version of the specification property
that is well adapted to the study of symbolic spaces with a non-uniform
structure.
