  We present the results of an analysis aimed at probing the small-scale
magnetic fields of M dwarfs observed with SPIRou, the nIR high-resolution
spectro-polarimeter installed at the Canada-France-Hawaii Telescope, in the
context of the SPIRou Legacy Survey. Our analysis relies on high-resolution
median spectra built from several tens of spectra recorded between 2019 and
2022, and on synthetic spectra computed with the ZeeTurbo code for various
combination of atmospheric parameters and magnetic field strengths. We pursue
the efforts undertaken in a previous study and focus on 44 weakly to moderately
active M dwarfs. We derive average magnetic field strengths (<$B$>) ranging
from 0.05 to 1.15 kG, in good agreement with activity estimates and rotation
periods. We found that including magnetic fields in our models has virtually no
impact on our derived atmospheric parameters, and that a priori assumptions on
the stellar surface gravity can affect our estimated <$B$>. Our results suggest
that small-scale magnetic fields account for more than 70% of the overall
average magnetic field for most targets whose large-scale fields were
previously measured. We derived low magnetic fluxes for several targets in our
sample, and found no clear evidence that <$B$> decreases with increasing Rossby
number in the unsaturated dynamo regime. We even identified counterexamples (GJ
1289 and GJ 1286) where the small-scale field is unusually strong despite the
long rotation period. Along with similar results on the large-scale fields, our
findings further suggest that dynamo processes may operate in a
non-conventional mode in these strongly magnetic, slowly-rotating stars.
