The magnetocaloric effect (MCE) was measured to probe the nature of phase coexistence of structurally different ferromagnetic metallic (FMM) and charge-ordered insulating phases in La5/8-yPryCa3/8MnO3 (y = 0.275) single crystals. The MCE peaks with both positive and negative values are observed in the vicinity of the multiple-phase transitions in the system. Strain associated with the phase coexistence has been known to stabilize a strain-glass state as well as a strain-liquid state. The large MCE is observed in the "dynamic" strain liquid state, while it is relatively small in the "frozen" strain-glass state. The MCE data reveal that the sharp increase in the magnetization below the Curie temperature in the strain-liquid region is attributed to the enhancement of the FM domain regions that are already present in the material. MCE is also shown to be a useful method to probe the subtle balance of coexisting phases in mixed-phase manganites.