Recently, we reported a new set of hyperbolic cosine type basis sets, which include the radial part of exponential type functions. In this study, it is show that the double hyperbolic cosine basis sets with non-integer Slater type orbitals give extremely accurate results, with the accuracy superior to that of the previous similar calculations. Total energy differences and comparison between double hyperbolic cosine and other similar basis sets suggested in literature are presented. Using double hyperbolic cosine orbitals, combined Hartree-Fock-Roothaan calculations have been carried out on the ground states of atoms and their ions within the minimal basis sets framework to compare the performance of the basis sets. The basis sets achieved the accuracy far beyond the double-zeta quality. Proposed basis sets can also play an effective role in ion-atom collisions problems and semi-empirical methods.