Transient stability analysis is a crucial tool for evaluating stability and ensuring safe operation of power systems. Among existing methodologies for transient stability analysis, direct methods show merits in performing fast contingency screening and providing quantitative information for the degree of stability. However, the inherent conservatism of direct methods and their restriction on power system models still pose significant challenges for practical applications. This paper is devoted to further developing direct methods by proposing a novel adaptive Lyapunov function method , which enables estimation of critical clearing time with drastically reduced conservatism. The novelties of the proposed method lie in three aspects. First, we propose an adaptive sector condition which bounds the nonlinearity of the power system model in an adjustable neighborhood of a given equilibrium point. Second, we introduce an improved bounding technique for the time derivative of the Lyapunov function. Third, by exploiting the freedom of the adaptive sector condition and the adjustable neighborhood , the construction of Lyapunov functions along with the choice of the parameters in the sector conditions can be co-optimized to achieve the tightest possible estimation of the critical clearing time. The effectiveness of the proposed method is validated on four benchmark systems.