A computationally efficient direct higher-order Taylor-Galerkin method is presented. A predictor-corrector type higher-order Taylor-Galerkin method has been developed by the first author. However, the method, which is highly accurate compared to other existing methods, is conditionally stable. For dynamic analysis of complicated structures other than trusses, it is essential for an algorithm to be unconditionally stable. In this work, an unconditionally stable scheme is presented. This scheme preserves the desirable properties of the Taylor-Galerkin method in that it requires neither a special starting algorithm nor the evaluation of acceleration at each time step. Along with the unconditionally stable scheme, an accurate but conditionally stable scheme is also presented. Accuracy and computational efficiency of the schemes are discussed.