Abstract We explore a new, efficient mechanism that can power toroidally magnetized jets up to two to three times their original terminal velocity after they enter a self-similar phase of magnetic acceleration. Underneath the elongated outflow lobe formed by a magnetized bubble, a wide-angle free wind, through the interplay with its ambient toroid, is compressed and accelerated around its axial jet. The extremely magnetic bubble can inflate over its original size, depending on the initial Alfvén Mach number M A of the launched flow. The shape-independent slope ∂ v r /∂ r = 2/3 t is a salient feature of the self-similarity in the acceleration phase. Peculiar kinematic signatures are observable in the PV diagrams and can combine with other morphological signatures as probes for density-collimated jets arising in toroidally dominated magnetized winds. The apparent second acceleration is powered by the decrease of the toroidal magnetic field but operates far beyond the scales of the primary magnetocentrifugal launch region and the free asymptotic terminal state. Rich implications may connect the jets arising from the youngest protostellar outflows such as HH 211 and HH 212 and similar systems with parsec-scale jets across the mass and evolutionary spectra.