In the last few years, progressively more frequent and precise direct detections of gravitational waves have facilitated sensitive tests of general relativity. Among the most intriguing possibilities is the existence of horizonless objects more compact than the photon ring. If such objects exist, it is inevitable that in addition to a standard ringdown, there will be late-time echoes of gravitational waves. Searches for echoes have so far been inconclusive, but here we propose more sensitive investigations that incorporate stacking of multiple events. We also suggest model-independent search techniques to add robustness and rigor to explorations of the presence of echoes in gravitational wave data.