A Broyden-Fletcher-Goldfarb-Shanno (BFGS) update formula is a standard technique for updating the Hessian matrix of a Lagrangian function in a sequential quadratic programming (SQP). The initial Hessian of the SQP is usually set to an identity matrix, because the previous information of the Hessian does not exist at the first iteration and it is extremely expensive to evaluate the exact Hessian of the real Lagrangian function. The inaccuracy of the identity matrix, however, is propagated to the next iterations in the SQP using BFGS update formula. In this study, we develop a new method that can generate more accurate approximate Hessian than that using the BFGS update formula even if the identity matrix is employed at the first iteration. In this method, the inaccuracy of the identity matrix is not propagated to the next iterations. Since the approximate Lagrangian obtained by using an enhanced two-point diagonal quadratic approximation method can be expressed as an explicit function of the design variables, the Hessian of the approximate Lagrangian can be analytically evaluated with negligible computational cost.