In this work, we develop an hp-adaptivity strategy for the minimum action method (MAM) using a posteriori error estimate. MAM plays an important role in minimizing the Freidlin-Wentzell action functional, which is the central object of the Freidlin-Wentzell theory of large deviations for noise-induced transitions in stochastic dynamical systems. Because of the demanding computation cost, especially in spatially extended systems, numerical efficiency is a critical issue for MAM. Difficulties come from both temporal and spatial discretizations. One severe hurdle for the application of MAM to large scale systems is the global reparametrization in time direction, which is needed in most versions of MAM to achieve accuracy. We recently introduced a new version of MAM in [22], called tMAM, where we used some simple heuristic criteria to demonstrate that tMAM can be effectively coupled with $h$-adaptivity, i.e., the global reparametrization can be removed. The target of this paper is to integrate $hp$-adaptivity into tMAM using a posteriori error estimation techniques, which provides a general adaptive MAM more suitable for parallel computing. More specifically, we use the zero-Hamiltonian constraint to define an indicator to measure the error induced by linear time scaling, and the derivative recovery technique to construct an error indicator and a regularity indicator for the transition paths approximated by finite elements. Strategies for $hp$-adaptivity have been developed. Numerical results are presented.