The inverse Faraday effect (IFE), which usually refers to the phenomenon in which a quasi-static axial magnetic field is self-generated when a circularly polarized beam propagates in a plasma, has rarely been studied for lasers with unconventional polarization states. In this paper, IFE is reconsidered for weakly relativistic full Poincaré beams, which can contain all possible laser polarization states. Starting from cold electron fluid equations and the conservation of generalized vorticity, a self-consistent theoretical model combining the nonlinear azimuthal current and diamagnetic current is presented. The theoretical results show that when such a laser propagates in a plasma, an azimuthally varying quasi-static axial magnetic field can be generated, which is quite different from the circularly polarized case. These results are qualitatively and quantitatively verified by three-dimensional particle-in-cell simulations. Our work extends the theoretical understanding of the IFE and provides a new degree of freedom in the design of magnetized plasma devices.