The existence of primary shearless tori is a distinguishing feature of nontwist maps. However, secondary shearless tori have been identified in the phase space of twist maps, in the neighbourhood of peculiar bifurcations of elliptic fixed points. In this paper, we report secondary shearless bifurcations in a twist symplectic map that describes chaotic field lines in tokamaks with an ergodic limiter. We identify the onset of secondary shearless tori, around elliptic fixed points within the island of stability, by examining numerical profiles of the internal rotation number. We present examples of field line transport barriers, associated with secondary shearless tori and their rupture, which reduce the usual magnetic field line escape at the tokamak plasma edge.