Helical core fibers (HCFs) suffer from low coupling efficiency and unavoidable excitation of higher order modes below the cutoff wavelength because of a core tilt with respect to the symmetry axis of the cladding. We propose an effective way of increasing the coupling efficiency to a HCF by untwisting its beginning section in a hydrogen flame. The proposed solution provides also a control over the excitation of higher order modes in HCFs and can be applied in splicing as well as in a free-space launching configuration. We experimentally demonstrate that by using the proposed method, the coupling efficiency between a SMF-28 and HCF can be increased to the level reachable for straight fibers, which is limited only by a modal fields mismatch. We also present detailed numerical and analytical studies of the coupling efficiency between a HCF and SMF versus the pitch distance in the partially untwisted input section of the HCF, which takes into account fundamental and higher-order modes.