The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.