We propose a stable sixth-order compact finite difference scheme coupled with a fifth-order staggered boundary scheme and the Runge-Kutta adaptive time stepping based on 3(2) Bogacki-Shampine pairs for pricing American options. To compute the free-boundary simultaneously and precisely with the option value and Greeks, we introduce a logarithmic Landau transformation and then remove the convective term in the pricing model by introducing the delta sensitivity, so that an efficient sixth-order compact scheme can be easily implemented. The main challenge in coupling the sixth order compact scheme in discrete form is to efficiently account for the near-boundary scheme. In this study, we introduce novel fifth and sixth-order Dirichlet near-boundary schemes that are suitable for solving our model. The optimal exercise boundary and other boundary values are approximated using a high-order analytical approximation that is obtained from a novel fifth-order staggered boundary scheme. Furthermore, we investigate the smoothness of the first and second-order derivatives of the optimal exercise boundary which is obtained from the high-order analytical approximation. Coupled with the adaptive time integration method, the interior values are then approximated using the sixth order compact schemes. As such, the expected convergence rate is reasonably achieved, and the present numerical scheme is very fast in computation and gives highly accurate solutions with very coarse grids.