Here we present experimental results on laser-driven ion acceleration from relativistically transparent, overdense plasmas in the break-out afterburner (BOA) regime. Experiments were preformed at the Trident ultra-high contrast laser facility at Los Alamos National Laboratory, and at the Texas Petawatt laser facility, located in the University of Texas at Austin. It is shown that when the target becomes relativistically transparent to the laser, an epoch of dramatic acceleration of ions occurs that lasts until the electron density in the expanding target reduces to the critical density in the non-relativistic limit. For given laser parameters, the optimal target thickness yielding the highest maximum ion energy is one in which this time window for ion acceleration overlaps with the intensity peak of the laser pulse. A simple analytic model of relativistically induced transparency is presented for plasma expansion at the