Semiconductor lasers subject to delayed optical feedback have recently shown great potential in solving computationally hard tasks. By optically implementing a neuro-inspired computational scheme, called reservoir computing, based on the transient response to optical data injection, high processing speeds have been demonstrated. While previous efforts have focused on signal bandwidths limited by the semiconductor laser's relaxation oscillation frequency, we demonstrate numerically that the much faster phase response makes significantly higher processing speeds attainable. Moreover, this also leads to shorter external cavity lengths facilitating future on-chip implementations. We numerically benchmark our system on a chaotic time-series prediction task considering two different feedback configurations. The results show that a prediction error below 4% can be obtained when the data is processed at 0.25 GSamples/s. In addition, our insight into the phase dynamics of optical injection in a semiconductor laser also provides a clear understanding of the system performance at different pump current levels, even below solitary laser threshold. Considering spontaneous emission noise and noise in the readout layer, we obtain good prediction performance at fast processing speeds for realistic values of the noise strength.