  We present new results of our ongoing project toward a precision
determination of the kaon $B$ parameter with the Kogut-Susskind quark action in
quenched QCD. New results taken at $\beta$=6.4 and $\beta=5.7$ suggest that an
apparently linear $a$ dependence of $B_K$ previously observed for
$\beta=5.85-6.2$ arises from a change of curvature from convex to concave as
the lattice spacing is reduced. Fitting data for $\beta\geq 5.93$ with an
$O(a^2)$ form yields $B_K$(NDR,2 GeV)=0.587(7)(17) in the continuum limit. We
also describe a finite-size study of $B_K$ at $\beta=$ 6.0 and 6.4, and a
reanalysis of the theoretical argument for $O(a^2)$ behavior.
