The $mu$eV axion is a well-motivated extension to the standard model. The Axion Dark Matter eXperiment (ADMX) collaboration seeks to discover this particle by looking for the resonant conversion of dark-matter axions to microwave photons in a strong magnetic field. In this Letter, we report results from a pathfinder experiment, the ADMX Sidecar, which is designed to pave the way for future, higher mass, searches. This testbed experiment lives inside of and operates in tandem with the main ADMX experiment. The Sidecar experiment excludes masses in three widely spaced frequency ranges (4202-4249, 5086-5799, and 7173-7203 MHz). In addition, Sidecar demonstrates the successful use of a piezoelectric actuator for cavity tuning. Finally, this publication is the first to report data measured using both the TM$_{010}$ and TM$_{020}$ modes.
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