Parton model contributions as next-to-eikonal corrections to the dipole factorization of DIS and SIDIS at low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:

We compute the next-to-eikonal (NEik) power corrections to inclusive deep inelastic scattering (DIS) and semi-inclusive deep inelastic scattering (SIDIS) at low <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> </a:math> beyond dipole factorization, which represent the eikonal result. The analysis is restricted to contributions arising from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>t</c:mi> </c:math> -channel quark exchanges, thereby probing the quark background field of the target. For a transversely polarized virtual photon, the NEik corrections to inclusive DIS are expressed in terms of quark and antiquark collinear parton distribution functions, while the corresponding corrections to SIDIS are formulated in terms of the unpolarized quark transverse-momentum-dependent distribution. In contrast, for a longitudinally polarized photon, the NEik corrections to both inclusive DIS and SIDIS vanish at lowest order in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>α</e:mi> <e:mi>s</e:mi> </e:msub> </e:math> .