This paper proposes multi-timescale voltage stability-constrained volt/VAR optimization to improve the steady-state stability and security of smart distribution grids. It aims to coordinate the post-contingency slow-timescale corrective actions provided by an on-load tap changer with fast-timescale corrective actions provided by photovoltaic inverters and battery energy storage system. This multi-timescale operation exploits the difference between short- and long-term emergency ratings of line flows and voltage magnitudes usually defined by utilities for contingency cases. Global optimum is achieved by formulating the problem as mixed-integer second-order cone program. To justify the investment in battery storage, the proper coordination between base case state-of-charge and storage dynamic reserve margin required for post-contingency corrective actions is performed. Also, the steady-state voltage stability margin is computed through detailed voltage stability analysis. Simulation results demonstrate the effectiveness of the proposed multi-timescale corrective actions in maintaining the steady-state stability and security under contingency cases.