  The decay $D\to K^-\pi^+$ is studied in a sample of quantum-correlated
$D\bar{D}$ pairs, based on a data set corresponding to an integrated luminosity
of 2.93\,fb$^{-1}$ collected at the $\psi(3770)$ resonance by the BESIII
experiment. The asymmetry between $C\!P$-odd and $C\!P$-even eigenstate decays
into $K^-\pi^+$ is determined to be ${\cal A}_{K\pi} = 0.132 \pm 0.011 \pm
0.007$, where the first uncertainty is statistical and the second is
systematic. This measurement is an update of an earlier study exploiting
additional tagging modes, including several decay modes involving a $K^0_L$
meson. The branching fractions of the $K^0_L$ modes are determined as input to
the analysis in a manner that is independent of any strong phase uncertainty.
Using the predominantly $C\!P$-even tag $D\to \pi^+\pi^-\pi^0$ and the ensemble
of $C\!P$-odd eigenstate tags, the observable ${\cal A}_{K\pi}^{\pi\pi\pi^0}$
is measured to be $0.130 \pm 0.012 \pm 0.008$. The two asymmetries are
sensitive to $r_D^{K\pi} \cos \delta_D^{K\pi}$, where $r_D^{K\pi}$ and
$\delta_D^{K\pi}$ are the ratio of amplitudes and phase difference,
respectively, between the doubly Cabibbo-suppressed and Cabibbo-favoured
decays. In addition, events containing $D \to K^-\pi^+$ tagged by $D \to
K^0_{S,L} \pi^+\pi^-$ are studied in bins of phase space of the three-body
decays. This analysis has sensitivity to both $r_D^{K\pi} \cos \delta_D^{K\pi}$
and $r_D^{K\pi} \sin \delta_D^{K\pi}$. A fit to ${\cal A}_{K\pi}$, ${\cal
A}_{K\pi}^{\pi\pi\pi^0}$ and the phase-space distribution of the $D \to
K^0_{S,L} \pi^+\pi^-$ tags yields $\delta_D^{K\pi}= \left( 187.6
{^{+8.9}_{-9.7}}{^{+5.4}_{-6.4}} \right)$ degrees, where external constraints
are applied for $r_D^{K\pi}$ and other relevant parameters. This is the most
precise measurement of $\delta_D^{K\pi}$ in quantum-correlated $D\bar{D}$
decays.
