This paper studies the coordinated direct and relay transmission (CDRT) in the non-orthogonal multiple access network, which consists of a base station (BS), a cell-center user (CCU), a cell edge user (CEU), and a group of half-duplex relays. During each transmission, the BS sends information to the CCU directly and to the CEU with the help of one of the relays. To improve the transmission reliability, a novel dynamic detection CDRT (DD-CDRT) strategy is designed, which fully exploits the available side information to eliminate the inter-user interference. Based on the DD-CDRT strategy, we further propose a relay selection scheme which maximizes the CCU's successful decoding probability with guaranteeing the reliable reception of the CEU. To evaluate our designs, exact and asymptotic outage probabilities are derived for each user. The results show that the achievable diversity order for the CEU is equal to the number of relays. Meanwhile, the CCU achieves a diversity order as one. Specifically, when the system sum rate is relatively small or the relay number is large enough, the outage performance of the CCU can approach the performance of the interference-free situation. Finally, the numerical results are presented to verify the theoretical analysis and to demonstrate the advantages of the DD-CDRT strategy using the proposed relay selection scheme.