Cyclic m-diagonal matrices possessing a Toeplitz structure have been extensively studied in the contemporary literature. In this work, we propose two efficient numerical procedures for calculating the determinant of general cyclic m-diagonal matrices possessing a Toeplitz structure. The first procedure is based on converting the matrix into a 2-by-2 block matrix format, subsequently leveraging the inverse of this reformulated matrix. The second technique is based on a conversion to an upper triangular block matrix. The performance of these new algorithms is then benchmarked against established methods from prior research.