We report on a bifurcation scenario from which emerges oscillations at a frequency higher than the relaxationoscillation frequency in an edge-emitting laser diode subjected to polarized optical feedback; these oscillations appear in the square-wave regime. The study has been performed experimentally and numerically based on Lang-Kobayashi model. We unveil bifurcation diagrams showing a clear bifurcation point that marks the transition between sustained and damped oscillations on the plateaus of square-waves. We extend this first bifurcation study by providing additional analytical insight to characterize the main parametric dependence of the frequency of these undamped oscillations. © 2018 SPIE.