  We demonstrate theoretically that the golden phase of SmS ($g$-SmS), a
correlated mixed-valent system, exhibits nontrivial surface states with diverse
topology. It turns out that this material is an ideal playground to investigate
different band topologies in different surface terminations. We have explored
surface states on three different (001), (111), and (110) surface terminations.
Topological signature in the (001) surface is not apparent due to a hidden
Dirac cone inside the bulk-projected bands. In contrast, the (111) surface
shows a clear gapless Dirac cone in the gap region, demonstrating the
unambiguous topological Kondo nature of $g$-SmS. Most interestingly, the (110)
surface exhibits both topological-insulator-type and
topological-crystalline-insulator (TCI)-type surface states simultaneously. Two
different types of double Dirac cones, Rashba-type and TCI-type, realized on
the (001) and (110) surfaces, respectively, are analyzed with the mirror
eigenvalues and mirror Chern numbers obtained from the model-independent
\emph{ab initio} band calculations.
