High critical current density (J(c)) values in superconducting wires/tapes are desirable for high magnetic field applications. Recently developed pnictide wires/tapes exhibit exceptional superconducting properties such as high critical temperature (T-c), upper critical field (H-c2), and almost field-independent J(c). Despite the great fabrication efforts, however, the newly discovered pnictidewires/tapes are still not able to replace low-temperature superconductors such as Nb3Sn, due to their inferior J(c) values. Ag-clad Sr0.6K0.4Fe2As2 tapes have demonstrated significant superconducting performance, although their low J(c) in comparison to Nb3Sn is still a major challenge. By successfully employing hydrostatic pressure, a remarkably significant enhancement of J(c) by an order of magnitude can be achieved in Sr0.6K0.4Fe2As2 tapes in both low and high fields. This is a promising technological step forward towards high-field applications, as the record high J(c) values (similar to 2 x 10(5) A/cm(2) at 4.2 K and 13 T, P = 1.1GPa) obtained for Sr0.6K0.4Fe2As2 tape are superior to those of Nb3Sn and other pnictide wires/tapes. Here, we used magnetic J(c) data for comparison to the other reported transport J(c) data, due to the lack of transport measurement facility under hydrostatic pressure. Our systematic analysis shows that pressure-induced pinning centers are the main source of J(c) enhancement, along with a fractional contribution from geometric changes around the grain boundaries under pressure. We expect that utilization of an appropriate pressure approach will be a way to significantly enhance J(c) to beyond the cutoff (maximum) values in various superconductors produced using other existing methods for J(c) enhancement.