Although diffusion properties of a suspension of swimming microorganisms in equilibrium have been studied intensively, those under nonequilibrium conditions remain unclear. In this study, we investigate the spreading of model microorganisms from high concentration to low by the Stokesian dynamics method. The results reveal that the spreading is neither purely diffusive nor ballistic. When the dipole component of the swimming velocity is small, the cells actively direct themselves towards lower concentrations. The concentration distribution shows stronger oscillations than would be expected for ballistic swimmers with constant orientations. The mechanism can be explained by the near-field hydrodynamic interactions between cells and the spatial gradient of the collision rate. Comparison of the numerical results with a simple continuum model and a Monte Carlo simulation shows that those conventional models can capture the basic features of the present results. These new findings pave the way towards a mathematical description of the dispersion of microorganisms in various environments.