In this paper, we perform analytical and numerical studies of TEM-wave propagation in an elliptic coaxial waveguide filled with a graded inhomogeneous RHM-LHM composite. Within the composite, the real parts of the effective permittivity and permeability vary along the propagation direction (chosen to be the z-direction) according to a hyperbolic tangent function. The present model introduces the transition width as an additional degree of freedom in the graded metamaterial composite design, which allows for tailoring of the electromagnetic wave propagation. An excellent agreement is obtained between the analytical results and numerical results obtained using the COMSOL Multiphysics software.