  We re-examine the combined semi-classical and mean-field limit in the
$N$-body fermionic Schr\"odinger equation with pure state initial data using
the Husimi measure framework. The Husimi measure equation involves three
residue types: kinetic, semiclassical, and mean-field. The main result of this
paper is to provide better estimates for the kinetic and mean-field residue
than those in \cite{Chen2021JSP}. Especially, the estimate for the mean-field
residue is shown to be smaller than the semiclassical residue by a mixed-norm
estimate of the two-particle reduced density matrix factorization. Our analysis
also updates the oscillation estimate parts in the residual term estimates
appeared in \cite{Chen2021JSP}.
