This work presents an adaptive superfast proximal augmented Lagrangian (AS-PAL) method for solving linearly-constrained smooth nonconvex composite optimization problems. At each iteration, AS-PAL inexactly solves a possibly nonconvex proximal augmented Lagrangian subproblem with prox stepsize chosen aggressively large so as to speed up its termination. An adaptive ACG variant of FISTA, namely R-FISTA, is then used to possibly solve the above subproblem. AS-PAL then adaptively updates the prox stepsizes based off the possible failures of R-FISTA. An interesting feature of AS-PAL is that it does not require knowledge of the parameters (e.g., size of constraint matrix, objective function curvatures, etc) underlying the problem. We demonstrate the speed and efficiency of our method through extensive computational experiments which show that AS-PAL can be more than 10 times faster than all the other state-ofthe-art codes on many instances (particularly when high accuracy is required).