This paper utilizes a high-performance aircraft to carry a single-sphere neutron spectrometer to measure the cosmic-ray atmospheric neutron energy spectra at different flight altitudes. The single-sphere neutron spectrometer employs a polyethylene sphere with a diameter of 29.5 cm as the moderator to thermalize neutrons of different energies. Neutron detection is achieved using 36 6Li-7Li glass scintillators distributed at various depths within the polyethylene moderating sphere. The 6Li glass scintillator responds to both neutron and gamma signals, while the 7Li glass scintillator is used to measure the gamma background signal. By subtracting the gamma background from the mixed neutron-gamma signal detected by the 6Li glass scintillator, the neutron counts can be obtained. A 1 cm thick lead shell embedded within the polyethylene moderating sphere extends the neutron energy response of the detector up to 1 GeV, enabling the measured neutron energy spectrum to cover the range from thermal neutrons to the GeV region. Compared to the Bonner multi-sphere spectrometer, the single-sphere neutron spectrometer significantly reduces volume and weight, meeting the requirements for airborne measurements. The neutron response function of the single-sphere neutron spectrometer detector was obtained using FLUKA simulations, and the atmospheric neutron energy spectrum can be derived from the detector neutron counts based on the unfolding algorithm. Flight-based experiments were conducted on the Suining–Luoyang and Suining–Changsha routes, obtaining measured data of the atmospheric neutron energy spectrum in China's airspace at altitudes of 4.2-7.6 km. The neutron energy spectra at different altitudes exhibit similar shapes, consisting of four energy regions: a thermal neutron peak below 0.4 eV, an epithermal neutron plateau region between 0.5 eV and 0.1 MeV, an "evaporation" peak between 0.1 MeV and 20 MeV, and a cascade peak above 20 MeV. The accuracy of the measurement results was verified by comparing the measured atmospheric neutron energy spectra with the calculated values from the EXcel-based Program for calculating Atmospheric Cosmic-ray Spectrum (EXPACS) model. Based on the measured atmospheric neutron energy spectra, the fluxes of thermal neutrons and high-energy neutrons, which contribute primarily to single-event effects in semiconductor devices, can be obtained, with a ratio of approximately 0.76 between the two. As the flight altitude increases from 4.2 km to 7.6 km, the high-energy neutron flux above 10 MeV increases from 117 n·cm^-12·h^-1 to 358 n·cm^-12·h^-1. With increasing altitude, the high-energy neutron flux approximately follows an exponential growth trend. These results provide critical data support for the analysis of atmospheric neutron-induced failure risks in semiconductor devices.