A free-boundary formulation is considered for the price of American options under jump-diffusion models with finite jump activity. On the free boundary a Cauchy boundary condition holds, due to the smoothpasting principle. An implicit finite difference discretization is performed on time-dependent non-uniform grids. During time stepping, solutions are interpolated from one grid to another, using Lagrange interpolations. Finite difference stencils are also constructed, using Lagrange interpolation polynomials, based on either three or five grid points. With these choices, second-order and fourth-order convergence with respect to the number of time and space steps can be expected. In numerical examples these convergence rates are observed under the Black-Scholes model and Kou's jump-diffusion model.