We describe the application of displaced, or misaligned, beams in a mirror-based magneto-optical trap (MOT) to enable portable and miniaturized atom chip experiments where optical access is limited to a single window. Two different geometries of beam displacement are investigated: a variation on the well-known ‘vortex-MOT’, and the other a novel ‘hybrid-MOT’ combining Zeeman-shifted and purely optical scattering force components. The beam geometry is obtained similar to the mirror-MOT, using a planar mirror surface but with a different magnetic field geometry more suited to planar systems. Using these techniques, we have trapped around 6 × 106 and 26 × 106 atoms of 85Rb in the vortex-MOT and hybrid-MOT respectively. For the vortex-MOT the atoms are directly cooled well below the Doppler temperature without any additional sub-Doppler cooling stage, whereas the temperature of the hybrid-MOT has been measured slightly above the Doppler temperature limit. In both cases the attained lower temperature ensures the quantum behaviour of the trapped atoms required for the applications of portable quantum sensors and many others.