The linewidth enhancement factor α is a key parameter determining the spectral and dynamical behavior of semiconductor lasers. Here, we propose and demonstrate a method for determining this parameter based on a direct measurement of variations in the laser gain and emission spectrum when subject to delayed optical feedback. We then use our approach to determine the pump current dependent linewidth enhancement factor of a high-β quantum dot micropillar laser. The validity of our approach is confirmed comparing it to two conventional methods, one based on the comparison of the linewidths above and below threshold and the other based on injection locking properties. Furthermore, the pump power dependence of α is quantitatively described by simulations based on a quantum-optical model.