Aims. We aim to use the high spatial resolution of the Atacama Large Millimeter/submillimeter Array (ALMA) to map the flow pattern of molecular gas near DG Tauri and its disk, a young stellar object driving a jet and a molecular outflow. Methods. We use observations from ALMA in the J = 2 − 1 transition of ¹²CO, ¹³CO, and C¹⁸O to study the Keplerian disk of DG Tauri and outflows that may be related to the disk and the jet. Results. We find a new wind component flowing radially at a steep angle (≈25° from the vertical) above the disk with a velocity of ≈3.1 km s⁻¹. It continues the trend of decreasing velocity for increasing distance from the jet axis (“onion-like velocity structure”). Conclusions. The new component is located close to the protostellar disk surface and may be related to photoevaporative winds.