In this article, we study the physical-layer security (PHY-security) improvement method for a downlink nonorthogonal multiple access (NOMA) system in the presence of an active eavesdropper. To this end, we propose a full-duplex (FD)-enabled NOMA system and a promising scheme, called the minimal transmitter selection (MTS) scheme, to support secure transmission. Specifically, the cell-center and cell-edge users act simultaneously as both receivers and jammers to degrade the eavesdropper channel condition. Additionally, the proposed MTS scheme opportunistically selects the transmitter to minimize the maximum eavesdropper channel capacity. To estimate the secrecy performance of the proposed methods, we derive an approximated closed-form expression for secrecy outage probability (SOP) and build a deep neural network (DNN) model for SOP evaluation. Numerical results reveal that the proposed NOMA system and MTS scheme improve not only the SOP but also the secrecy sum throughput. Furthermore, the estimated SOP through the DNN model is shown to be tightly close to other approaches, i.e., the Monte-Carlo method and analytical expressions. The advantages and drawbacks of the proposed transmitter selection scheme are highlighted, along with insightful discussions.