<p>This paper addresses the interval overestimation problem inherent in the extra unitary interval method, which stems from neglected variable dependencies during multiplication operations. An improved extra unitary interval algorithm is proposed to resolve this issue by accurately estimating higher-order terms from known lower-order information, thereby effectively mitigating dependency-related overestimation and significantly enhancing the precision of interval analysis. An adaptive second-order interval finite element method using the improved extra unitary interval is further developed. The proposed method first employs a full-term second-order Taylor expansion to accurately approximate the stiffness matrix and load vector, thereby capturing essential nonlinear effects. Then, the Neumann series is utilized to approximate the inverse of the stiffness matrix and further obtain the explicit expression of the displacement response vector. Finally, the improved extra unitary interval algorithm is utilized to compute the displacement bounds, while an adaptive mechanism ensures solution convergence with minimal computational cost. Numerical examples of three engineering structures, subject to varying uncertainty levels, demonstrate the method's effectiveness and accuracy.</p>