Owing to the flexible mobility and convenient deployment, unmanned aerial vehicle (UAV) aided networks can facilitate many applications. However, the open accessibility of wireless communication brings a great risk of privacy to UAV-based networks. Thus, in this paper, we propose a UAV-aided covert communication scheme assisted by a multi-antenna jammer to maximize the transmission rate between a ground transmitter and a UAV receiver against several randomly distributed wardens. The transmitter adopts the maximum ratio transmission, while the jammer zero-forces its transmitted signal at the UAV to disturb the monitoring at wardens without interfering the legitimate transmission. First, we analyze the detection performance and derive the optimal threshold for each warden to minimize its detection outage probability (DOP). Then, with the worst situation in which all wardens set their respective optimal thresholds to achieve the minimum global DOP, the location and the transmit power of the jammer are optimized to maximize the DOP. The location of UAV and the transmit power of the ground transmitter are also optimized to maximize the transmission rate with the minimum DOP requirement satisfied. Numerical results are provided to demonstrate the effectiveness of the proposed UAV-aided covert communication scheme.