Optimal controllers for time-delay systems are inherently infinite-dimensional. To implement such designs on practical systems, these controllers must be approximated by finite-dimensional models that preserve stability and minimize performance degradation. This problem, known as the best H-infinity approximation, has been a longstanding challenge due to the uniqueness of the optimal solution and the lack of an explicit analytical formula. Various approximation techniques, such as LMI-based methods, nonlinear optimization, the iterative rational Krylov algorithm, and moment-matching techniques, attempt to achieve sub-optimal solutions. However, the resulting approximation error is highly sensitive to initial pole selection, or the choice of significant frequency components considered during the approximation process. This work introduces a direct method for determining significant frequencies associated with peak errors in the equioscillating behavior of the best H-infinity approximation. The approach is closely related with the Hankel norm approximation technique and based on Trefethen’s extension of the Chebyshev Equioscillation theorem for rational approximations. Specifically, the method utilizes the poles of the Hankel approximation with one additional order and the residual anti-stable part obtained by the eigenvalue decomposition, to determine these critical frequencies. By using these frequencies, the proposed approach improves initialization strategies for state-of-the-art techniques, enhancing the accuracy of the approximation. Various numerical examples are shown by using different methods to demonstrate the effectiveness of the proposed approach. In addition, this study is part of an ongoing effort to develop non-iterative techniques for sub-optimal H-infinity approximations, aiming to establish more computationally efficient solutions for finite dimensional controller design.