Context. Non-dusty late-type giants without a corona and large-scale pulsations represent objects that do not fulfil the conditions under which standard mass-loss mechanisms can be applied efficiently. Despite the progress during the past decades, the driving mechanism of their winds is still unknown. Aims. One of the crucial constraints of aspiring wind-driving theories can be provided by the measured velocity and density fields of outflowing matter. The main goal of this work is to match the radial velocities of absorbing matter with a depth in the red giant (RG) atmosphere in the S-type symbiotic star EG And. Methods. We measured fluxes and radial velocities of ten Fe I absorption lines from spectroscopic observations with a resolution of ≈30 000. At selected orbital phases, we modelled their broadened profiles, including all significant broadening mechanisms. Results. The selected Fe I absorption lines at 5151–6469 Å originate at a radial distance ≈1.03 RG radii from its centre. The corresponding radial velocity is typically ≈1 km s−1, which represents a few percent of the terminal velocity of the RG wind. The high scatter of the radial velocities of several km s−1 in the narrow layer of the stellar atmosphere points to the complex nature of the near-surface wind mass flow. The average rotational velocity of 11 km s−1 implies that the rotation of the donor star can contribute to observed focusing the wind towards the orbital plane. The orbital variability of the absorbed flux indicates the highest column densities of the wind in the area between the binary components, even though the absorbing neutral material is geometrically more extended from the opposite side of the giant. This wind density asymmetry in the orbital plane region can be ascribed to gravitational focusing by the white dwarf companion. Conclusions. Our results suggest that both gravitational and rotational focusing contribute to the observed enhancement of the RG wind towards the orbital plane, which makes mass transfer by the stellar wind highly efficient.