The outputs of two independent, incoherent lasers are spatially overlapped in a Raman-active medium. When the pulses are separated in time, the stimulated Raman scattering (SRS) efficiency of the delayed pulse is increased from 14 to 27%. We show that this occurs because of the presence of the decaying coherent excitation induced by SRS of the first pulse. The influence of saturable losses is removed by prepulsing the medium prior to applying the pulse pair. The temporal evolution of the enhancement and phase-matching conditions are investigated. Surprisingly, the pulses interact only when they are separated in time, with the enhancement vanishing when they are temporally overlapped. These results show that the outputs of several independent lasers may be combined to generate significantly higher SRS efficiencies, without increasing the pump intensity incident on the medium. This novel technique provides a means for recovering efficient SRS in the transient (short-pulse) regime.