  We present results for the kaon mixing parameter $B_K$ obtained using
improved staggered fermions on a much enlarged set of MILC asqtad lattices.
Compared to our previous publication, which was based largely on a single
ensemble at each of the three lattice spacings $a\approx 0.09\;$fm, $0.06\;$fm
and $0.045\;$fm, we have added seven new fine and four new superfine ensembles,
with a range of values of the light and strange sea-quark masses. We have also
increased the number of measurements on one of the original ensembles. This
allows us to do controlled extrapolations in the light and strange sea-quark
masses, which we do simultaneously with the continuum extrapolation. This
reduces the extrapolation error and improves the reliability of our error
estimates. Our final result is $\hat{B}_K = 0.7379 \pm 0.0047 (\text{stat}) \pm
0.0365 (\text{sys})$.
