Using a relativistic electromagnetic particle code, we investigate positron acceleration in a shock wave propagating obliquely to an external magnetic field ${\bf B} _0 $ in an electron–positron–ion plasma. After an encounter with a shock wave, some positrons are reflected and then accelerated along the magnetic field. They stay in the shock transition region and have velocities nearly parallel to ${\bf B} _0 $. Owing to the deformation of the wave profile, the acceleration can become stagnant. However, if the shock speed $v_{\rm sh}$ is close to $c \cos \theta $, where $c$ is the speed of light and $\theta $ is the angle between the wave normal and ${\bf B} _0 $, the acceleration can start again. In this second stage acceleration, three types of motion are found. The first type is the same as that in the first stage. In the second type, particles gain energy from the electric field perpendicular to the magnetic field in association with large-radius gyromotion. The third type of motion resembles the curtate cycloid in the wave frame. We give theoretical estimates for the energy increase in these processes.