We study an inverse resonance problem on a radially and compactly perturbed infinite hyperbolic cylinder. Using the symmetries of this kind of geometry, we are led to study a stationary Schrödinger equation on the line with a potential V which is the sum of a Pöschl-Teller potential and a perturbation which we consider integrable and compactly supported. We define the resonances as the poles of the reflection coefficients with a negative imaginary part. We prove that, under some assumptions on the support of the compact perturbation, we are able to solve the question of uniqueness in the inverse resonance problem. We also give asymptotics of the resonances and show that they are asymptotically localised on two logarithmic branches and according to the localisation of the support of q, sometimes also on parallel lines to the imaginary axis.