  We advance the threshold resummation formalism for semi-inclusive
deep-inelastic scattering (SIDIS) to next-to-next-to-next-to-leading
logarithmic (N$^{3}$LL) order, including the three-loop hard factor. We expand
the results in the strong coupling to obtain approximate
next-to-next-to-next-to-leading order (N$^{3}$LO) corrections for the SIDIS
cross section. In Mellin moment space, these corrections include all terms that
are logarithmically enhanced at threshold, or that are constant. We also
consider a set of corrections that are suppressed near threshold. Our numerical
estimates show modest changes of the cross section by the approximate N$^{3}$LO
terms, suggesting a very good perturbative stability of the SIDIS process.
