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It is broadly believed that quasinormal modes (QNMs) cannot tell theblack-hole near-horizon geometry, because usually the low-lying modes aredetermined by the scattering of perturbations around the peak of the effectivepotential. Using the general parametrization of the black-hole spacetimesrespecting the generic post-Newtonian asymptotic, we will show that tinymodifications of the Schwarzschild/Kerr geometry in a small region near theevent horizon lead to almost the same Schwarzschild/Kerr fundamental mode, buttotally different first few overtones. Having in mind that the first severalovertones affect the quasinormal (QN) ringing at its early and intermediatestage [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 studiedvia the first few overtones of the QN spectrum, which is important becausecorrections to the Einstein theory must modify precisely the near-horizongeometry, keeping the known weak field regime. We discuss the connection ofthis observation with the so called "overtones' instability" recently studiedin [J. Jaramillo et. al. Phys. Rev. Lett. 128, 211102 (2022)].

First few overtones probe the event horizon geometry