  Within the Color Glass Condensate (CGC) effective field theory, we derive the
next-to-leading order (NLO) cross-section for the single-jet semi-inclusive
cross-section in deep inelastic scattering (DIS) at small $x$, for both
longitudinally and transversely polarized virtual photons. We provide analytic
expressions, valid at finite $N_c$ and suitable for numerical evaluation, for
both the cross-section differential in rapidity and transverse momentum and the
cross-section differential in rapidity only. Our NLO formulae demonstrate that
the very forward rapidity regime is plagued by large double logarithmic
corrections coming from phase space constraints on soft gluons close to the
kinematic threshold for jet production. A joint resummation of small-$x$ and
threshold logarithms at single logarithmic accuracy is proposed to remedy the
instability of the cross-section in this regime. By integrating over the
single-jet phase space, we recover known results for the NLO DIS structure
functions at small $x$, previously obtained using the optical theorem.
