It is common practice to truncate the finite-volume formula for $K\to\pi\pi$, and other one-to-two transitions, to only include the lowest partial wave, as in the original derivation by Lellouch and Lüscher. However, as the precision of lattice calculations increases, it may become important to assess the systematic uncertainty of this approximation. With this motivation, we compare the $S$-wave-only ($\ell=0$) results with those truncated at the next lowest value of angular momentum that can contribute.