The goal of this paper is to introduce a novel likelihood-based inferential framework for axion haloscopes which is valid under the commonly applied "rescanning" protocol. The proposed method enjoys short data acquisition times and a simple tuning of the detector configuration. Local statistical significance and power are computed analytically, avoiding the need of burdensome simulations. Adequate corrections for the look-elsewhere effect are also discussed. The performance of our inferential strategy is compared with that of a simple method which exploits the geometric probability of rescan. Finally, we exemplify the method with an application to a HAYSTAC type axion haloscope.

14 pages, 12 figures