  In experiments searching for axionic dark matter, the use of the standard
threshold-based data analysis discards valuable information. We present a
Bayesian analysis framework that builds on an existing processing protocol to
extract more information from the data of coherent axion detectors such as
operating haloscopes. The analysis avoids logical subtleties that accompany the
standard analysis framework and enables greater experimental flexibility on
future data runs. Performing this analysis on the existing data from the
HAYSTAC experiment, we find improved constraints on the axion-photon coupling
$g_\gamma$ while also identifying the most promising regions of parameter space
within the $23.15$--$24.0$ $\mu$eV mass range. A comparison with the standard
threshold analysis suggests a $36\%$ improvement in scan rate from our
analysis, demonstrating the utility of this framework for future axion
haloscope analyses.
