Abstract: In this paper, a new hybrid augmented finite volume method is proposed for solving porous media equations. Based on the idea of the constant variation method, a robust method of Puiseux series expansion with augmented variables is successfully constructed to recover the case of complete nonlinear degenerate diffusion coefficients. By "seamlessly coupling" the singular subdomain and the regular subdomain by augmented variables, the stationary porous media equation on the regular subdomain can be solved by using the augmented finite volume scheme with uniform mesh. It is worth noting that compared with traditional numerical methods, this method shows significant advantages in solving one-dimensional and two-dimensional stationary porous media equations. The results show that for the one-dimensional stationary porous media equations with different parameter values, the proposed method not only maintains the correctness of numerical solutions, but also exhibits unique precision advantages and robustness, especially when dealing with challenges such as coefficient singularities and low regularity of the solution space. Two-dimensional numerical experiments show that the method has the potential of high-dimensional applications.