  The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of
any contractible periodic Reeb orbit of a non-degenerate toric contact form on
a good toric contact manifold with zero first Chern class, i.e. a Gorenstein
toric contact manifold, can be explicitly computed using moment map data. In
this paper we show that the same explicit method can be used to compute
Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under
appropriate conditions, the (finite) number of such orbits in a given free
homotopy class and with a given index is a contact invariant of the underlying
contact manifold. We compute these invariants for two sets of examples that
satisfy these conditions: $5$-dimensional contact manifolds that arise as unit
cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensional
Gorenstein contact lens spaces. As applications, we will see that these
invariants can be used to show that diffeomorphic lens spaces might not be
contactomorphic and that there are homotopy classes of diffeomorphisms of some
lens spaces that do not contain any contactomorphism. Following a suggestion by
one referee, we will also see that this type of applications can be proved
alternatively by looking at the total Chern class of these canonical contact
structures on lens spaces.
