The competition between the Kondo correlation and superconductivity in quantum-dot Josephson junctions (QDJJs) has been known to drive a quantum phase transition between 0 and π junctions. Theoretical studies so far have predicted that under strong Coulomb correlations the 0−π transition should go through intermediate states, 0′ and π′ phases. By combining a nonperturbative numerical method and the resistively shunted
junction model, we investigated the magnetic-field-driven phase transition of the QDJJs in the Kondo regime and found that the low-field magnetotransport exhibits a unique feature which can be used to distinguish the intermediate phases. In particular, the magnetic-field driven π′−π transition is found to lead to the enhancement of the supercurrent which is strongly related to the Kondo effect.