We describe advances on a method designed to derive accurate parameters of M dwarfs. Our analysis consists in comparing high-resolution infrared spectra acquired with the near-infrared spectro-polarimeter SPIRou to synthetic spectra computed from MARCS model atmospheres, in order to derive the effective temperature (Teff), surface gravity (log g), metallicity ($\rm {[M/H]}$), and alpha-enhancement ($\rm {[\alpha /Fe]}$) of 44 M dwarfs monitored within the SPIRou Legacy Survey (SLS). Relying on 12 of these stars, we calibrated our method by refining our selection of well-modelled stellar lines, and adjusted the line list parameters to improve the fit when necessary. Our retrieved Teff, log g, and $\rm {[M/H]}$ are in good agreement with literature values, with dispersions of the order of 50 K in Teff and 0.1 dex in log g and $\rm {[M/H]}$. We report that fitting $\rm {[\alpha /Fe]}$ has an impact on the derivation of the other stellar parameters, motivating us to extend our fitting procedure to this additional parameter. We find that our retrieved $\rm {[\alpha /Fe]}$ are compatible with those expected from empirical relations derived in other studies.