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In this paper, we provide new formulas for determining the coefficientsappearing in the asymptotic expansion for the Barnes $G$-function as $n$ tendsto infinity for certain classes of asymptotic expansion for the Barnes$G$-function. We remark that our formulas can be used to approximate thecoefficients appearing in an asymptotic expansion of the ``random matrixfactor" from the Keathing-Snaith conjecture and the coefficients appearing inan asymptotic expansion of the ``L\'evy-Khintchine type representation of thereciprocal of the Barnes $G$-function".

New approximations for the higher order coefficients in an asymptotic expansion for the Barnes $G$-function