  It is broadly believed that quasinormal modes (QNMs) cannot tell the
black-hole near-horizon geometry, because usually the low-lying modes are
determined by the scattering of perturbations around the peak of the effective
potential. Using the general parametrization of the black-hole spacetimes
respecting the generic post-Newtonian asymptotic, we will show that tiny
modifications of the Schwarzschild/Kerr geometry in a small region near the
event horizon lead to almost the same Schwarzschild/Kerr fundamental mode, but
totally different first few overtones. Having in mind that the first several
overtones affect the quasinormal (QN) ringing at its early and intermediate
stage [M. Giesler, M. Isi, M. Scheel, and S. Teukolsky, Phys. Rev. X 9, 041060
(2019)], we argue that the near-horizon geometry could in principle be studied
via the first few overtones of the QN spectrum, which is important because
corrections to the Einstein theory must modify precisely the near-horizon
geometry, keeping the known weak field regime. We discuss the connection of
this observation with the so called "overtones' instability" recently studied
in [J. Jaramillo et. al. Phys. Rev. Lett. 128, 211102 (2022)].
