We examine the impact of several factors on electron acceleration by a laser pulse and the resulting electron energy gain. Specifically, we consider the role played by: (1) static longitudinal electric field, (2) static transverse electric field, (3) electron injection into the laser pulse, and (4) static longitudinal magnetic field. It is shown that all of these factors lead, under certain conditions, to a considerable electron energy gain from the laser pulse. In contrast with other mechanisms such as wakefield acceleration, the static electric fields in this case do not directly transfer substantial energy to the electron. Instead, they reduce the longitudinal dephasing between the electron and the laser beam, which then allows the electron to gain extra energy from the beam. The mechanisms discussed here are relevant to experiments with under-dense gas jets, as well as to experiments with solid-density targets involving an extended pre-plasma.