We report a comprehensive high-pressure study on the triple-layer T'-La4Ni3O8 with a suite of experimental probes, including structure determination, magnetic, and transport properties up to 50 GPa. Consistent with a recent ab inito calculation, application of hydrostatic pressure suppresses an insulator-metal spin-state transition at P(c)≈6 GPa. However, a low-spin metallic phase does not emerge after the high-spin state is suppressed to the lowest temperature. For P>20 GPa, the ambient T' structure transforms gradually to a T(†)-type structure, which involves a structural reconstruction from fluorite La-O2-La blocks under low pressures to rock-salt LaO-LaO blocks under high pressures. Absence of the metallic phase under pressure has been discussed in terms of local displacements of O2- ions in the fluorite block under pressure before a global T(†) phase is established.