The dynamics of a BCS superconductor driven out-of-equilibrium by a laser pulse is analyzed in great detail on varying the temperature and the laser pulse characteristics with respect to the maximum gap amplitude. The observed effects include: oscillations both in the amplitude and in the phase of the superconducting order parameter, suppression of the superconductivity, but also its enhancement with a strong dependence on all varying parameters. The microscopic understanding of these effects is obtained by studying thoroughly the evolution of the amplitude and the phase of the momentum-dependent components of the gap and of the momentum-resolved occupation of the system. The latter, although remains constant in sum overall, also evolve in time following the variations caused by the applied field in the dispersion. This study can be quite relevant to investigate the different behavior of nodal and anti-nodal thermal and non-thermal excitations.