We report experimental bifurcation diagrams of a semiconductor laser, biased well above threshold, subjected to external optical feedback. As feedback is increased, we see a quasiperiodic route to chaos interrupted by several windows of periodicity corresponding to limit cycles, differing in frequency by multiples of the external-cavity free-spectral range that have developed around external-cavity modes (ECMs) whose frequency is slightly larger than that of the solitary laser. Successive windows correspond to the transition between two limit cycles either on the same or neighboring ECMs. For larger feedback, the laser operates in a chaotic regime around numerous negatively shifted external-cavity modes. These experimental observations detail the bifurcations leading to fully developed chaos in this system, and further provide detailed insight on the standard theoretical framework applied to these lasers.