  We discuss the subleading power corrections to one-jet production processes
in $N$-jettiness subtraction using vector-boson plus jet production as an
example. We analytically derive the next-to-leading power leading logarithmic
corrections (NLP-LL) through ${\cal O}(\alpha_S)$ in perturbative QCD, and
outline the calculation of the next-to-leading logarithmic corrections
(NLP-NLL). Our result is differential in the jet transverse momentum and
rapidity, and in the vector boson momentum squared and rapidity. We present
simple formulae that separate the NLP corrections into universal factors valid
for any one-jet cross section and process-dependent matrix-element corrections.
We discuss in detail features of the NLP corrections such as the process
independence of the leading-logarithmic result that occurs due to the
factorization of matrix elements in the subleading soft limit, the occurrence
of poles in the non-hemisphere soft function at NLP and the cancellation of
potential $\sqrt{\mathcal{T}_1/Q}$ corrections to the $N$-jettiness
factorization theorem. We validate our analytic result by comparing them to
numerically-fitted coefficients, finding good agreement for both the inclusive
and the differential cross sections.
