This paper introduced a new aftertreatment technique for solving a fractional nonlinear Susceptible-Infectious-Recovered (SIR)-epidemic model. The proposed approach reformulated the power series solution via incorporating the Laplace transform and its inverse. Basically, it applied the Laplace transform to convert the series solution into different Pad $\acute{e}$-approximants involving Laplace's parameter with arbitrary order. This procedure facilitated the method of deriving the inverse Laplace transform explicitly, as a final step, using basic special functions in fractional calculus. Our analysis was capable of obtaining a sequence of closed-form approximations in terms of the Mittag-Leffler functions. The current results may be provided for the first time regarding the Mittag-Leffler solution of the fractional SIR-model. Additionally, the outcome of this analysis revealed that our approach was not only effective but also applicable to a wide range of fractional differential equations and systems.