  General relativity admits a plethora of exact compact object solutions. The
augmentation of Einstein's action with non-minimal coupling terms leads to
modified theories with rich structure, which, in turn, provide non-trivial
solutions with intriguing phenomenology. Thus, assessing their viability under
generic fluctuations is of utmost importance for gravity theories. We consider
static and spherically-symmetric solutions of a Horndeski subclass which
includes a massless scalar field non-minimally coupled to the Einstein tensor.
Such theory possesses second-order field equations and admits an exact black
hole solution with scalar hair. Here, we study the stability of such solution
under axial gravitational perturbations and find that it is linearly stable.
The qualitative features of the ringdown waveform depend solely on the ratio of
the two available parameters of spacetime, namely the black hole mass $m$ and
the non-minimal coupling strength $\ell_\eta$. Finally, we demonstrate the
gravitational-wave ringdown transitions between three distinct patterns as the
ratio $m/\ell_\eta$ increases; a state which is dominated by photon-sphere
excitations and maintains a typical quasinormal ringdown, an intermediate
long-lived state which exhibits gravitational-wave echoes and, finally, a state
where the ringdown and echoes are depleted rapidly to turn to an exponential
tail.
