We study the approximate channel simulation problem in the normal approximation regime. Our starting point is the fact established by Cao et al.~\cite{Cao_channel_simulation}, that the minimal codebook size $M_ε^*(W)$ for a fixed error tolerance $ε>0$ is closely related to the $ε$-smooth channel max-information, $I_{\max}^ε(W)$. Using this, we refine earlier large-blocklength approximations to $M_ε^*(W^{\otimes n})$ for discrete memoryless channels. Specifically, we provide tight third-order expansions for Gallager-symmetric channels and demonstrate that the third-order term is channel-dependent. As in corresponding channel coding expansions, we identify a substantial rate gap between singular and nonsingular channels. Notably, the direction of this gap is reversed relative to channel coding: The third-order term is $-\frac{1}{2}\log n$ for singular channels but $O(\log\log n)$ for nonsingular channels.

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