  We propose to extend the optimal Fermi averaging procedure theoretically in
order to calculate an in-medium $K^-n\to \pi^- \Lambda$ amplitude for the
exothermic ($K^-$, $\pi^-$) reaction on nuclei in the framework of a
distorted-wave impulse approximation, taking into account the local momentum
transfer generated by semiclassical distorted waves for $K^-$ and $\pi^-$
mesons. Angular distributions for the $^{12}$C($K^-$, $\pi^-$)$^{12}_\Lambda$C
reaction in which a momentum transfer is $q \lesssim$ 80 MeV/$c$ at $p_K=$ 800
MeV/$c$ in the $\pi^-$ forward direction, are estimated by applying the
extended procedure. The result shows that the calculated angular distributions
are in good agreement with those of the data, and this extension is a
successful prescription making it possible to describe the reaction cross
sections in near-recoilless reactions such as ($K^-$, $\pi^-$) in our
framework.
