We report on the results from a search for dark matter axions with the HAYSTAC experiment using a microwave cavity detector at frequencies between 5.6 and 5.8 GHz. We exclude axion models with two photon coupling gaγγ≳2×10−14 GeV−1, a factor of 2.7 above the benchmark KSVZ model over the mass range 23.15<ma<24.0 μeV. This doubles the range reported in our previous paper. We achieve a near-quantum-limited sensitivity by operating at a temperature T<hν/2kB and incorporating a Josephson parametric amplifier (JPA), with improvements in the cooling of the cavity further reducing the experiment’s system noise temperature to only twice the standard quantum limit at its operational frequency, an order of magnitude better than any other dark matter microwave cavity experiment to date. This result concludes the first phase of the HAYSTAC program utilizing a conventional copper cavity and a single JPA.