In order to couple a terahertz Gaussian beam into a hyperbolic metal waveguide(HMWG), a tapered elliptical-hyperbolic metal waveguide(TEHMWG) was proposed as a connector. The input of the TEHMWG was rectangular, and the output was elliptical-hyperbolic. In the tapered process, the TE01 mode of rectangular metal waveguide was gradually transformed into the elliptical polarized mode of hyperbolic metal waveguide. According to WKB approximation, the slowly tapered waveguide eliminated the reflection and scattering. The coupling efficiency between the TE01 mode and the elliptical polarized mode was 94%. Through this connector, the coupling efficiency between a Gaussian beam and HMWG was improved to 69.1%.