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Recently it was shown that small deformations of the asymptotically flat black-hole geometry in some region near its horizon which do not alter considerably the fundamental mode, nevertheless, strongly affect the first several (and higher) overtones which deviate at an increasing rate from their nondeformed limits. Here we show that, despite the quasinormal spectrum in anti-de Sitter (AdS) space is totally different from the asymptotically flat one, the outburst of overtones do take place at small near-horizon deformations of Schwarzschild-AdS spacetime as well. Moreover, qualitatively new, nonoscillatory modes appear as a result of such small deformations, representing, thereby, a nonperturbative branch of modes. For this purpose we extend the general parametrization of asymptotically flat black holes to the AdS case. Small near-horizon deformations may originate from various holographically motivated factors, such as quantum corrections or attempts to describe the regime of intermediate coupling via higher curvature terms, while the nonoscillatory modes may be related to the hydrodynamic mode on the gauge theory side. Therefore, the phenomenon of the overtones' outburst must be taken into consideration when analyzing correlation functions and dispersion relations in the dual field theory. In addition to the ad hoc deformations we consider the case of five-dimensional Einstein-Gauss-Bonnet-AdS black holes as an example of such near horizon deformations and fulfill the detailed study of the overtones.