Motivated by the recent experimental measurements of differential cross sections of the $Sigma^{-}p$ elastic scattering in the momentum range of $470$ to $850$ MeV$/c$ by the J-PARC E$40$ experiment, we extend our previous studies of $S=-1$ hyperon-nucleon interactions to relatively higher energies up to $900$ MeV$/c$ for both the coupled-channel $Lambda prightarrow(Lambda p, Sigma^{+}n, Sigma^{0}p)$, $Sigma^{-}prightarrow(Lambda n, Sigma^{0}n, Sigma^{-}p)$ and single-channel $Sigma^{+}prightarrowSigma^{+}p$ reactions. We show that although the leading order covariant chiral effective field theory is only constrained by the low energy data, it can describe the high energy data very well, in particular, the J-PARC E40 differential cross sections. In particular, we predict a pronounced cusp structure close to the $Sigma N$ threshold in the $Lambda pto Lambda p$ reaction, which can be checked in the future using, e.g., the Femtoscopy technique. The predicted total and differential cross sections are of relevance for ongoing and planned experiments.
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