  We revisit the calculation of quark and scalar propagators in a gluon
background field at Next-to-Eikonal accuracy, which will provide the first
power correction to scattering processes in the high-energy limit, including
gluon saturation effects. The two main results of our study are as follows.
First, we derive a very general Next-to-Eikonal expansion of the quark
propagator in terms of the scalar propagator, which allows us to isolate
spinor-specific Next-to-Eikonal corrections from the universal ones. Second, we
present a new method allowing, for the first time, a systematic calculation of
the Next-to-Eikonal corrections coming from the $x^-$ dependence of the
background field. This opens the possibility to study high-energy scattering
processes on a target with non-trivial dynamics instead of on a static one, by
going beyond infinite Lorentz time dilation of the target.
