Gravitational wave echoes from black holes have been suggested as a crucial observable to probe the spacetime in the vicinity of the horizon. In particular, it was speculated that the echoes are closely connected with specific characteristics of the exotic compact objects, and moreover, possibly provide an access to the quantum nature of gravity. Recently, it was shown that the discontinuity in the black hole metric substantially modifies the asymptotical behavior of quasinormal frequencies. In the present study, we proceed further and argue that a discontinuity planted into the metric furnishes an alternative mechanism for the black hole echoes. Physically, the latter may correspond to an uneven matter distribution inside the surrounding halo. To demonstrate the results, we first numerically investigate the temporal evolution of the scalar perturbations around a black hole that possesses a nonsmooth effective potential. It is shown that the phenomenon persists even though the discontinuity can be located further away from the horizon with rather insignificant strength. Besides, we show that the echoes in the present model can be derived analytically based on the modified pole structure of the associated Green function. The asymptotical properties of the quasinormal mode spectrum and the echoes are found to be closely connected, as both features can be attributed to the same origin. In particular, the period of the echoes in the time domain is shown to be related to the asymptotic spacing between successive poles along the real axis in the frequency domain Δ(ℜω) by a simple relation limℜω→+∞Δ(ℜω)=2π/T. Moreover, we discuss possible distinguishment between different echo mechanisms. The potential astrophysical implications of the present findings are also addressed.