  The latest search for the rare kaon decay $K^+\to\pi^+\nu\bar\nu$ by the NA62
experiment has produced evidence for it with a branching fraction consistent
with the prediction of the standard model. The new result implies that in this
decay, with the $\nu\bar\nu$ pair appearing as missing energy ($\not\!\!E$),
the room for possible new physics is no longer sizable and that therefore its
contributions to underlying four-particle $s\to d\!\not\!\!E$ operators with
parity-even $ds$ quark bilinears have become significantly constrained.
Nevertheless, we point out that appreciable manifestations from beyond the
standard model induced by the corresponding operators with mainly parity-odd
$ds$ quark bilinears could still occur in $K\to\pi\pi\!\not\!\!E$ modes, on
which there are only minimal empirical details at present. We find in
particular that new physics of this kind may enhance the branching fraction of
$K_L\to\pi^0\pi^0\!\not\!\!E$ to values reaching its current experimental upper
limit and the branching fractions of $K^+\to\pi^+\pi^0\!\not\!\!E$ and
$K_L\to\pi^+\pi^-\!\not\!\!E$ to the levels of $10^{-7}$ and $10^{-6}$,
respectively. Thus, quests for these decays in existing kaon facilities such as
KOTO and NA62 or future ones could provide valuable information complementary
to that gained from $K\to\pi\!\not\!\!E$.
