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 LISAs 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.
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