In this letter, the outage probability of an unmanned aerial vehicle (UAV) network with hybrid-duplex (HBD) UAV communications is investigated in a stochastic geometry framework. We demonstrate that the HBD UAV communication system (HBD-UCS) can concurrently support more UAVs while achieving higher reliability than the half-duplex (HD) UCS (HD-UCS). Specifically, at low transmit power regimes, it is shown that the HBD-UCS attains lower uplink and downlink outage probability than an HD-UCS, as the UAV operating altitude is increased.