We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp., parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge--Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyze stability and consistency, and illustrate with numerical results.