The purpose of this paper is to study the effect of conformal perturbations on the local smoothing effect for the Schrödinger equation on surfaces of revolution. The paper [CW13] studied the Schrödinger equation on surfaces of revolution with one trapped orbit. The dynamics near this trapping were unstable, but degenerately so. Beginning from the metric g from these papers, we consider the perturbed metric g_s = e^sf g, where f is a smooth, compactly supported function. If s is small enough and finitely many derivatives of f satisfy an appropriate bound, then we show that a local smoothing estimate still holds.