Liquid metals, like lithium (Li), are considered a promising plasma-facing material due to their self-repairing, in comparison with the conventional solid materials that have limitations in handling high heat flux in future fusion devices. To predictively simulate global Li transport under the lithium divertor condition, the three-dimensional Monte Carlo code ITCD has been upgraded significantly, in terms of the simulation domain (from the sole divertor region in a limited toroidal range to the entire edge plasma region in a full toroidal torus). The expansion of the simulation zone brings about the new demand of the computational resource, which motivates us to implement the guiding-center (GC) particle push approach into ITCD. The trajectory of charged Li particle using the GC particle push approach shows a good agreement with the full-orbit (FO) particle push method. The FO and GC hybrid particle push scheme has been used to deal with the gyration scrape-off effect and meanwhile speed up the calculation of the global Li transport. The characteristics of Li impurity density and deposition distributions are studied in detail by ITCD.