This paper proposes a methodology for enhancing the power system flexibility, which can respond properly to contingencies in real-time operations. The proposed approach introduces a unified power flow controller (UPFC) in a three-stage security-constrained optimal power flow (SCOPF). The pre- and post-contingency system operation states are divided into three stages including the base case, post-contingency short-term, and post-contingency long-term periods. The UPFC applications re-route active power flow and provide reactive power to mitigate overloads and voltage violations when line outages occur in power systems. UPFC is adopted as a fast-response corrective control device during the post-contingency short-term period, which is coordinated with the conventional slow-response corrective control system during the post-contingency long-term period. A convex approach is applied to reformulate the original nonlinear nonconvex SCOPF problem into a second-order cone programming (SOCP) problem. A two-level algorithm using Benders decomposition and sequential cone programming (SCP) is applied to solve the large-scale SOCP problem. An improved covering cut bundle (CCB) strategy is proposed to accelerate the convergence of the Benders decomposition algorithm. Numerical results show the effectiveness of the proposed model and its solution technique for enhancing the power system flexibility.