We study the ground state properties of the S=1/2 staggered Heisenberg-\GammaΓ honeycomb model under a magnetic field based on analytical and numerical methods. Our calculations show that the conventional zigzag and stripy phases are favored because of the staggered Heisenberg interaction away from the pure \GammaΓ limit. In our classical analysis, we find that the field induces a series of competing magnetic phases with relatively large unit cells in the region sandwiched between the two magnetic phases with long-range ordering. In the quantum treatment, these large magnetic unit cells are destabilized by strong quantum fluctuations that result in the stabilization of a gapless quantum spin liquid behavior. In a honeycomb \GammaΓ magnet, we disclose an intermediate-field gapless quantum spin liquid phase driven by a tilted field away from the out-of-plane direction only for a narrow region between the low-field zigzag and high-field fully polarized phases.