  In this work we analyze some judiciously chosen solutions of Kerr Black Holes
with Scalar Hair (KBHsSH) of special interest for Gravitational Wave (GW)
events originated from Extreme Mass Ratio Inspirals (EMRIs). Because of the
off-center distribution of energy density, these spacetimes are warped in such
a way that not all metric functions behave monotonically on the equatorial
plane as in the exterior region of Kerr black holes (KBHs). This has great
impact on the orbital parameters, which in turn affects the imprints on signals
descendant from EMRIs in a adiabatic evolution. By investigating circular obit
parameters, we unveil what qualitative features could be present in the signals
that are new and distinct compared to KBHs, and we evolve some inspirals by
employing the usual quadrupole formula approximation. We show that the
frequencies of the emitted signals behave nonmonotonically, i.e. they can
backward chirp, and for some particular cases they can become arbitrarily
small, falling below LISA's sensibility range. Finally, we present two sets of
waveforms produced by a noncircular EMRI in which the compact object (CO)
follows a type of geodesic motion typically present in spacetimes with a static
ring (SR), in which the compact object is periodically momentarily at rest.
