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This article generalizes the implementation of higher order corrections to state transition matrices during instantaneous reversals in hybrid dynamical systems impacting a discontinuity boundary transversally. A closed form expression for saltation terms in systems possessing a degree of smoothness zero is derived. The difference in flight times of two closely initiated trajectories in state space to the impacting surface has been estimated up to $\mathcal{O}(2)$. A comparison of the times of impact estimated with the first order approximation reveals that higher order corrections lead to a significant improvement of estimates. Next, two new algorithms to estimate the Lyapunov spectrum and Floquet multipliers for piecewise-smooth systems have been presented using the derived second order corrections. Stability analyses are subsequently carried out using the proposed framework for two representative cases {\it i.e.,} of a hard impact oscillator and a pair impact oscillator. It is established that the obtained Floquet multipliers and Lyapunov spectrum accurately predict the stability of the dynamical states, as validated by their corresponding bifurcation diagrams.