  For studying small-$x$ gluon saturation in forward dijet production in
high-energy dilute-dense collisions, the improved TMD (ITMD) factorization
formula was recently proposed. In the Color Glass Condensate (CGC) framework,
it represents the leading term of an expansion in inverse powers of the hard
scale. It contains the leading-twist TMD factorization formula relevant for
small gluon's transverse momentum $k_t$, but also incorporates an all-order
resummation of kinematical twists, resulting in a proper matching to
high-energy factorization at large $k_t$. In this paper, we evaluate the
accuracy of the ITMD formula quantitatively, for the case of quark dijet
production in high-energy proton-proton($p+p$) and proton-nucleus ($p+A$)
collisions at LHC energies. We do so by comparing the quark-antiquark azimuthal
angle $\Delta\phi$ distribution to that obtained with the CGC formula. For a
dijet with each quark momentum $p_t$ much larger than the target saturation
scale, $Q_s$, the ITMD formula is a good approximation to the CGC formula in a
wide range of azimuthal angle. It becomes less accurate as the jet $p_t$'s are
lowered, as expected, due to the presence of genuine higher-twists
contributions in the CGC framework, which represent multi-body scattering
effects absent in the ITMD formula. We find that, as the hard jet momenta are
lowered, the accuracy of ITMD start by deteriorating at small angles, in the
high-energy-factorization regime, while in the TMD regime near
$\Delta\phi=\pi$, very low values of $p_t$ are needed to see differences
between the CGC and the ITMD formula. In addition, the genuine twists
corrections to ITMD become visible for higher values of $p_t$ in $p+A$
collisions, compared to $p+p$ collisions, signaling that they are enhanced by
the target saturation scale.
