We demonstrate the efficacy of the stochastic LapH method to treat all-to-all quark propagation on a $N_\mathrm{f} = 2+1$ CLS ensemble with large linear spatial extent $L = 5.5 \, \mathrm{fm}$ and $m_\pi L = 6.2$. Employing interlace-16 dilution in Laplacian eigenvector space, we obtain the benchmark elastic isovector $p$-wave pion-pion scattering amplitude to good precision already on a relatively small number of gauge configurations. These results hold promise for multi-hadron spectroscopy at close-to-physical pion mass with exponential finite-volume effects under control.