We study the tunneling zone solutions of a one-dimensional electrostatic potential for the relativistic (Dirac to Klein-Gordon) wave equation when the incoming wave packet exhibits the possibility of being almost totally transmitted through the barrier. The transmission probabilities, the phase times and the dwell times for the proposed relativistic dynamics are obtained and the conditions for the occurrence of accelerated tunneling transmission are all quantified. We show that, in some limiting cases, the analytical difficulties that arise when the stationary phase method is employed for obtaining phase (traversal) tunneling times are all overcome. Lessons concerning the phenomenology of the relativistic tunneling suggest revealing insights into condensed-matter experiments using electrostatic barriers for which the accelerated tunneling effect can be observed.551125132