  The Effective Field Theory (EFT) of perturbations on an arbitrary background
geometry with a timelike scalar profile has been recently constructed in the
context of scalar-tensor theories. Unlike General Relativity, the regular
Hayward metric is realized as an exact background metric in the Effective Field
Theory with timelike scalar profile without resorting to special matter field,
such as nonlinear electrodynamics. The fundamental quasinormal mode for axial
graviational perturbations of this black hole has been considered recently with
the help of various methods. Here we make a further step in this direction and
find that, unlike the fundamental mode, a few first overtones deviate from
their Schwarzschild limit at a much higher rate. This outburst of overtones
occurs because the overtones are extremely sensitive to the least change of the
near-horizon geometry. The analytical formula for quasinormal modes is obtained
in the eikonal regime. In addition, we calculated grey-body factors and showed
that the regular Hayward black hole with a scalar hair has a smaller grey-body
factor than the Schwarzschild one. Integration of the wave-like equation in the
time-domain shows that the power-law tails, following the ring-down phase, are
indistinguishable from the Schwarzschild ones at late times.
