We report results of polarized small-angle neutron scattering (SANS) experiments at T = 10 K and 300 K for cubic Fe3O4 submicron-sized particles, where formation of a complex spin vortex is expected. Magnetic SANS intensities of magnetization components in the direction perpendicular and parallel to the magnetic field were observed to change significantly at around the coercivity. Magnetization components parallel to the field minimizes around the coercivity both at T = 10 K and 300 K, whereas those perpendicular to the magnetic field maximizes near the coercivity and the maximum value differ greatly, depending on the temperature. Based on results of micromagnetic simulations, the observed SANS intensities were interpreted as due to magnetic structural changes from a flower to a spin vortex state and gradual tilting of a vortex core from the external field to magnetocrystalline anisotropy axes at low fields.