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Post-inflationary reheating is a widely discussed mechanism for non-thermal production of dark matter (DM). In this scenario the momentum distribution of the produced DM particles is usually taken to be the one obtained at reheating, red-shifted at later times due to the expansion of the Universe. However, since in such a scenario both the DM and the standard model (SM) fields couple to the inflaton, the DM particles necessarily undergo self-scatterings, as well as elastic and inelastic scattering reactions with the SM bath, all of which proceed through $s-$channel or $t-$channel inflaton exchange. We compute the momentum distribution of the DM particles including the effect of these scatterings, and find that the distributions can be significantly altered, even though DM remains non-thermal throughout the cosmological evolution. We observe that if the inflaton dominantly couples to the SM Higgs boson through a renormalizable interaction, then reheating temperatures and inflaton masses at the TeV scale lead to a large effect from the scattering processes, with the DM-inflaton coupling constrained by the DM density. The scattering effects are found to be sensitive to the duration of the reheating process -- larger the duration, more momentum modes are filled at reheating, leading to an enhanced scattering probability. We also obtain the free-streaming length of such DM using the resulting non-thermal momentum distribution, which can be used to estimate the implications of the Lyman-$α$ constraints on the DM mass. It is observed that in the scenarios considered, including the scattering effects can reduce the DM average velocity at matter-radiation equality, and its free-streaming length, by upto a factor of $40$, thereby making the constraints on light DM produced in inflaton decay significantly weaker.