We investigate the scatter-free propagation of low-energy (1--5 keV) protons in the magnetosheath by following test particle trajectories in a model of the magnetosheath fields previously obtained from gasdynamic simulations. We concentrate on those ions energized by near-specular reflection at the quasi-perpendicular shock: the reflected-gyrating ions. Our results indicate that for the most common orientations of the interplanetary field, it is unlikely that such ions when scattered in pitch angle behind the near-perpendicular bow shock (&thgr;Bn≥80¿) can contribute to upstream field-aligned beams leaving the bow shock at shock-normal angles (&thgr;Bn) greater than 45¿. Reflected-gyrating ions similarly scattered behind the quasi-perpendicular shock (&thgr;Bn≥45¿) are more likely to contribute to such beams by leaking from the bow shock close to where they entered the magnetosheath.