Summary: In a recent paper, some authors revealed that a modified 13-moment system taking intrinsic heat fluxes as variables, instead of the heat fluxes along the coordinate vectors which are adopted in the classical Grad 13-moment system, attains some additional advantages than the classical Grad 13-moment system, particularly including that the equilibrium is turned to be the interior point of its hyperbolicity region. The modified 13-moment system was actually derived from the generalized Hermite expansion of the distribution function, where the anisotropy of Hermite expansion is specified by the full temperature tensor. We extend the method therein in this paper to high order of generalized Hermite expansion to derive arbitrary order moment systems, and propose a globally hyperbolic regularization to achieve locally well-posedness similar to the method mentioned above. Furthermore, the structure of the eigen-system of the coefficient matrix and all characteristic waves are fully clarified. The obtained systems provide a systematic class of hydrodynamic models as the refined version of Euler equations, which are gradually approaching the Boltzmann equation with increasing order of the expansion.