The simulation of chemical kinetics involving multiple scales constitutes a modeling challenge (from ordinary differential equations to Markov chains) and a computational challenge (multiple scales, large dynamical systems, time step restrictions). In this paper, we propose a new discrete stochastic simulation algorithm: the postprocessed second kind stabilized orthogonal τ-leap Runge-Kutta method (PSK-τ-ROCK). In the context of chemical kinetics, this method can be seen as a stabilization of Gillespie's explicit τ-leap algorithm combined with a postprocessor. The stabilized procedure allows to simulate problems with multiple scales (stiff), while the postprocessing procedure allows to approximate the invariant measure (e.g. mean and variance) of ergodic stochastic dynamical systems. We prove stability and accuracy of the PSK-τ-ROCK for a reference system. Numerical experiments illustrate the high reliability and efficiency of the scheme when compared to other τ-leap methods.