It is well known that environment affects galaxy evolution, which is broadly related to supermassive black hole (SMBH) growth. We investigate whether SMBH evolution also depends on host-galaxy local (sub-Mpc) and global (~1-10Mpc) environment. We construct the surface-density field (local environment) and cosmic web (global environment) in the Cosmic Evolution Survey (COSMOS) field at z=0.3-3.0. The environments in COSMOS range from the field to clusters (M_halo_~<1014M_{sun}_), covering the environments where ~99 per cent of galaxies in the Universe reside. We measure sample-averaged SMBH accretion rate ({mean}BHAR) from X-ray observations, and study its dependence on overdensity and cosmic-web environment at different redshifts while controlling for galaxy stellar mass (M_*_). Our results show that {mean}BHAR does not significantly depend on overdensity or cosmic-web environment once M_*_ is controlled, indicating that environment-related physical mechanisms (e.g. tidal interaction and ram-pressure stripping) might not significantly affect SMBH growth. We find that {mean}BBHAR is strongly related to host-galaxy M_*_, regardless of environment.