Spin transport was studied in a two-dimensional electron gas hosted in a wide GaAs quantum well occupying two subbands. Using space and time Kerr rotation microscopy to image drifting spin packets under an in-plane accelerating electric field, optical injection and detection of spin polarization were achieved in a pump-probe configuration. The experimental data exhibited high spin mobility and long spin lifetimes allowing us to obtain the spin-orbit fields as a function of the spin velocities. Surprisingly, above moderate electric fields of 0.4 V/cm with velocities higher than 2 μm/ns, we observed a dependence of both bulk and structure-related spin-orbit interactions on the velocity magnitude. A remarkable feature is the increase in the cubic Dresselhaus term to approximately half of the linear coupling when the velocity is raised to 10 μm/ns. In contrast, the Rashba coupling for both subbands decreases to about half of its value in the same range. These results yield new information on the application of drift models in spin-orbit fields and about limitations for the operation of spin transistors.