The description of kinetics of physical aging, namely the slow evolution of a glass thermodynamic state toward equilibrium, generally relies on the exclusive role of the main α relaxation. Here, we study the kinetics of physical aging over a wide temperature range in five small molecules interacting via van der Waals forces monitoring the time evolution of the glass enthalpic state. To this aim, we employ fast scanning calorimetry, which permits exploring a wide range of aging times. To challenge the role of the α relaxation in the description of physical aging, we employ a model-independent approach, based on the time to reach equilibrium, and a modified version of the single parameter aging model. The latter accounts for the non-linearity of aging making use of the so-called density scaling approach to describe the dependence of the α relaxation time on the glass thermodynamic state. We show that the α relaxation is generally adequate to describe aging at temperatures close to the glass transition and, for lower temperatures, the latest stages of equilibration. In contrast, at low aging temperatures, it fails to catch a wide portion of the time-dependent evolution of the glass thermodynamic state, which is found to be much faster than predicted considering only the α relaxation. Hence, our results and analysis provide compelling arguments that the description of glass equilibration under a wide range of aging conditions is conveyed by different molecular mechanisms, beyond the mere role of the α relaxation.