  Scalar-tensor theory predicts solutions to the gravitational field equations
which describe compact objects in the presence of a non-minimally coupled
scalar field to the Einstein tensor. These objects are black holes with scalar
hair and wormholes supporting scalar phantom matter. The evolution of test
fields in fixed asymptotically-flat backgrounds of exotic compact objects leads
to the formation of echoes in the ringdown signal, which designate the
existence of trapping regions close to the event horizon. Here, we consider
minimally-coupled test scalar fields propagating on compact object solutions of
the Horndeski action, which possess an effective cosmological constant, leading
to anti-de Sitter asymptotics, and show that echoes can form in the ringdown
waveform due to the entrapment of test fields between the photon sphere and the
effective asymptotic boundary. Although the presence of an event horizon leads
to the usual echoes with decaying amplitude, signifying modal stability of the
scalarized black hole considered, we find that test scalar fields propagating
on a scalarized wormhole solution give rise to echoes of constant and equal
amplitude to that of the initial ringdown, indicating the existence of normal
modes. Finally, we find that, near extremality, the test field exhibits a
concatenation of echoes; the primary ones are associated with the trapping
region between the photon sphere and the effective anti-de Sitter boundary
while the secondary ones are linked to the existence of a potential well at the
throat of the wormhole.
