We exploit both covert communication and friendly jamming to propose a friendly jamming-assisted covert communication and use it to doubly secure a large-scale device-to-device (D2D) network against eavesdroppers (i.e., wardens). The D2D transmitters defend against the wardens by: 1) hiding their transmissions with enhanced covert communication, and 2) leveraging friendly jamming to ensure information secrecy even if the D2D transmissions are detected. We model the combat between the wardens and the D2D network (the transmitters and the friendly jammers) as a two-stage Stackelberg game. Therein, the wardens are the followers at the lower stage aiming to minimize their detection errors, and the D2D network is the leader at the upper stage aiming to maximize its utility (in terms of link reliability and communication security) subject to the constraint on communication covertness. We apply stochastic geometry to model the network spatial configuration so as to conduct a system-level study. We develop a bi-level optimization algorithm to search for the equilibrium of the proposed Stackelberg game based on the successive convex approximation (SCA) method and Rosenbrock method. Numerical results reveal interesting insights. We observe that without the assistance from the jammers, it is difficult to achieve covert communication on D2D transmission. Moreover, we illustrate the advantages of the proposed friendly jamming-assisted covert communication by comparing it with the information-theoretical secrecy approach in terms of the secure communication probability and network utility.