For systems that are not observable at an equilibrium to be stabilized, output-feedback stabilization is considerably challenging. This paper solves this control problem for the case-study of a second-order system that is bilinear and affine, both in the input and the output, but it is unobservable at the target equilibrium. The case-study defines a class of non-uniformly observable systems and stems from automotivecontrol applications. Our main contribution is a novel observerbased hybrid controller that achieves asymptotic stabilization semiglobally. The controller relies on a switched observer that estimates the state, provided that the latter is 'kept away' from the singular equilibrium. To achieve both competing tasks, stabilization and estimation, the controller relies on the keen construction of a piecewise-constant, converging, output reference. Our main results are illustrated via numerical simulations on a meaningful example.