In this paper, we present an improved version of the simplified lattice Boltzmann method on non-uniform meshes. This method is based on the recently-proposed simplified lattice Boltzmann method (SLBM) without evolution of the distribution functions. In SLBM, the macroscopic variables, rather than the distribution functions, are directly updated. Therefore, SLBM calls for lower cost in virtual memories and can directly implement physical boundary conditions. However, one big issue in SLBM is the lattice uniformity, which is inherited from the standard LBM and this makes SLBM only applicable on uniform meshes. To further extend SLBM to non-uniform meshes, Lagrange interpolation algorithm is introduced in this paper to determine quantities at positions where the streaming process is initiated. The theoretical foundation of the interpolation process is that both the equilibrium part and the non-equilibrium part of the distribution functions are continuous in physical space. In practical implementation, the Lagrange interpolation polynomials can be computed and stored in advance, due to which little extra efforts are brought into the computation. Three numerical tests are conducted with good agreements to reference data, which validates the robustness of the present method and shows its potential for applications to non-uniform meshes with curved boundaries.