The ground state of double perovskite oxide La2CoMnO6 (LCMO) and how it is influenced by external pressure and antisite disorder are investigated systematically by first-principles calculations. We find, on the consideration of both the electron correlation and spin-orbital coupling effect, that the LCMO takes on insulating nature, yet is transformed to half metallicity once the external pressure is introduced. Such tuning is accompanied by a spin-state transition of Co(2+) from the high-spin state (t(5)2ge(2)g) to low-spin state (t(6)2ge(1)g) because of the enhancement of crystal-field splitting under pressure. Using mean-field approximation theory, Curie temperature of LCMO with Co(2+) being in low-spin state is predicted to be higher than that in high-spin state, which is attributed to the enhanced ferromagnetic double exchange interaction arising from the shrinkage of Co-O and Mn-O bonds as well as to the increase in bond angle of Co-OMn under pressure. We also find that antisite disorder in LCMO enables such transition from insulating to half-metallic state as well, which is associated with the spin-state transition of antisite Co from high to low state. It is proposed that the substitution of La(3+) for the rare-earth (RE) ions with smaller ionic radii could open up an avenue to induce a spin-state transition of Co, rendering thereby the RE2CoMnO6 a promising half-metallic material.