Silicon nitride (SiN) integrated optical waveguides have found a wide range of applications due to their low loss, broad wavelength transmission band and high nonlinearity. However, the large mode mismatch between the single-mode fiber and the SiN waveguide creates a challenge of fiber coupling to these waveguides. Here, we propose a coupling approach between fiber and SiN waveguides by utilizing the high-index doped silica glass (HDSG) waveguide as the intermediary to smooth out the mode transition. We achieved fiber-to-SiN waveguide coupling efficiency of lower than 0.8 dB/facet across the full C and L bands with high fabrication and alignment tolerances.