The linewidth enhancement factor (LEF) is an important parameter of semiconductor lasers as it describes many of their unique properties. Recently, it is gaining broad attention due to its key role in frequency comb operation. However, previously developed techniques to measure the LEF are limited to sub-threshold bias or singlemode operation. Here, we report a novel and universally applicable technique to directly obtain the LEF of a running multimode laser frequency comb over its entire spectrum. The technique utilizes a phase-sensitive single shot measurement scheme. We derive a theoretical model, perform numerical Maxwell-Bloch simulations and experimentally demonstrate our technique on a quantum cascade laser.