A hybrid-duplex aeronautical communication system (HBD-ACS) consisting of a full-duplex (FD) enabled ground station (GS), and two half-duplex (HD) air-stations (ASs) is proposed as a direct solution to the spectrum crunch faced by the aviation industry. Closed-form outage probability and finite signal-to-noise ratio (SNR) diversity gain expressions in aeronautical communications over Rician fading channels are derived for a successive interference cancellation (SIC) detector. Similar expressions are also presented for an interference ignorant (II) detector and HD-equivalent modes at GS and ASs. Through outage and finite SNR diversity gain analysis conducted at the nodes, and system level, residual SI and inter-AS interference are found to be the primary limiting factors in the proposed HBD-ACS. Additional analysis also revealed that the II and SIC detectors in the proposed HBD-ACS are suitable for weak and strong interference scenarios, respectively. When compared to HD-ACS, the proposed HBD-ACS achieves lower outage probability and higher diversity gains at higher multiplexing gains when operating at low SNRs. Finite SNR analysis also showed the possibility of the proposed HBD-ACS being able to attain interference-free diversity gains through proper management of residual SI. Hence, the proposed HBD-ACS is more reliable and can provide better throughput compared to existing HD-ACS at low-to-moderate SNRs.