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 Tau 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 Tau 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 deg from the vertical) above the disk with a velocity of ~ 3.1 km/s. 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.