The nature of dark matter is one of the most outstanding mysteries in science today. Possibilities for its composition range from black holes to heavy particles that can be detected by their collisions with atomic nuclei to light particles that act like a coherent field. Sensitive experiments have not yet been able to detect dark matter. A case can be made that the most promising dark matter candidate is the axion, which was invented to solve the strong charge-parity problem in the standard model of particle physics. If the axion exists, it could be produced in abundance and constitute a major component of dark matter.In the presence of a magnetic field, axions will be converted to an oscillating electromagnetic field with a frequency approximately corresponding to the axion mass. We can detect this electromagnetic field as a power excess spectrally coincident with the resonance of a microwave cavity. Each microwave cavity can provide good sensitivity in the search for axions over a limited frequency range, but we must consider new technologies to probe the full axion parameter space in a reasonable amount of time. Both as a testbed for innovative cavity and amplifier concepts and as a data pathfinder for higher frequencies, the Haloscope at Yale Sensitive to Axion Cold dark matter (HAYSTAC) was the first cavity experiment to use a dilution refrigerator and commission a squeezed-state receiver to circumvent the Standard Quantum Limit. The HAYSTAC Phase I and II cavity is a cylinder with an internal rod that rotates off-center and tunes the mode-of-interest resonance frequency between 3.4 and 5.8 GHz in a volume of 1.5 L. Recent theoretical work favors axions between 4 and 12 GHz. I examined various cavity designs to access higher frequencies in this range and determined that a seven-rod cavity design would be optimal with a volume of 1.7 L for a frequency range between approximately 5.5 and 7.5 GHz. I designed this seven-rod cavity, constructed it, and characterized it in detail for use on HAYSTAC. My cavity design allows access to this well-motivated axion mass range with high sensitivity by increasing accessible frequencies without sacrificing performance. This seven-rod concept can be extended to higher frequencies.