Within the framework of the single-band effective mass approximation method, the Floquet theorem, and the transfer-matrix technique, we investigate single-electron photon-assisted tunnelling in a double-well potential with the time-periodic field and Rashba and Dresselhaus spin-orbit coupling. The transmission probability displays statellite peaks on both sides of the field-free resonant peaks. The results show that the single-electron spin tunnelling can be controlled through changing the structure of the double-quantum-well and the intensity of the applied electric field. These advantages are useful for optimizing the semiconductor spintronic devices.