We investigate the in-plane field dependence of current-induced magnetic domain wall motion in asymmetric Co/Ninano wires. The application of an in-plane field changed the direction of domain wall motion. In a systematic study, by varying the magnetic layer thickness, we show that the in-plane field induces a crossover from spin Hall torque to adiabatic spin-transfer torque for the magnetic domain wall motion. Furthermore, the dependence of the threshold current density on the in-plane longitudinal field provides a way for determining the strength of the Dzyaloshinskii-Moriya interaction.