A low‐beta tandem‐mirror trapped‐particle instability theory is developed for arbitrary azimuthal mode number including the effects of radial equilibrium electric fields. A stability window is seen to exist for inward‐pointing electric fields even when passing electrons bounce beyond passing ions. This stability window can persist at zero radial electric field when the instability drive is sufficiently small. The theory also predicts the possibility of an additional instability whenE×Brotation is present: a rotationally driven trapped‐particle mode.