We study axial gravitational perturbations of a hairy black hole constructed within the framework of gravitational decoupling and investigate the geometric origin of echo-like late-time signals in this spacetime. We derive the odd-parity master equation and the corresponding effective potential, and compute the quasinormal mode spectrum using frequency-domain and time-domain methods. We show that the axial potential develops a double-peak structure that supports a trapping cavity and gives rise to echo-like waveforms. The representative double-peak configurations also exhibit an unstable–stable–unstable light-ring structure, which provides the null-geodesic counterpart of the trapping cavity in the eikonal limit. Therefore, the echoes are not produced by an artificially imposed near-horizon condition, but instead emerge dynamically from the geometric structure of the effective potential. Our results provide a useful framework for probing black hole hair through gravitational-wave ringdown and for exploring possible observational departures from the standard no-hair paradigm.