An applied stress field sigma(3) can reversibly change the temperature of an elastocaloric material under adiabatic conditions, and the temperature change Delta T-sigma 3 is usually maximized near phase transitions. Using a thermodynamic approach, we demonstrate that an elastocaloric strength alpha = vertical bar Delta T-sigma 3 vertical bar/vertical bar sigma(3)vertical bar of 0.016 K/MPa can be achieved benefiting from the full first-order phase transition in BaTiO3 single crystals, which is comparable with typical elastocaloric materials reported in the literature. The elastocaloric temperature change is found to be giant (3.2 K) under a stress of 200 MPa with a temperature span of over 50 K, which can be significantly larger than its electrocaloric counterpart (similar to 1 K). Moreover, it is found that the elastocaloric strength can be remarkably enhanced (2.32 K/MPa) as long as the phase transition is triggered even by a modest stress near the sharp first-order phase transition, which is two orders of magnitude larger than those accomplished by full transition. Therefore, even a low stress (< 30 MPa) can induce a modest elastocaloric effect (1.3 K) comparable with the electrocaloric counterpart, which is accompanied by a reduction of the working temperature span. In addition, it is found that the electrocaloric peak under tensile stresses moves towards higher temperatures with its magnitude slightly enhanced. Hopefully, our study will stimulate further investigations on elastocaloric and stress-mediated electrocaloric effects in ferroelectrics.