Non-orthogonal multiple access (NOMA)-based cooperative relaying systems (CRS) has emerged as a solution to the spectral inefficiency problem of the conventional cooperative relaying systems thanks to the NOMA integration. Thus, as a subset of NOMA-CRS, the NOMA-based diamond relaying network (NOMA-DRN) also provides a performance gain in terms of throughput. However, the NOMA-DRN has a poor error performance due to the second phase (uplink), indeed, it has a error floor regardless of the transmit power, power allocation and channel qualities. To address this problem, in this paper, we propose a novel NOMA-DRN schemes where joint maximum likelihood (JML) decoding is implemented at the destination. Then, we define the performance metrics (i.e., bit error rate (BER) and the diversity order ) of the NOMA-DRN with the JML and analyze the computational complexity. Moreover, we demonstrate that the new NOMA-DRN with JML can cope with the error floor penalty of the conventional NOMA-DRN. Hence, a spectral efficient NOMA-CRS scheme can be achieved with high data reliability. Specifically, this improvement can reach to $\sim20-30dB$ in the transmit power which is superb gain in terms of energy efficiency perspective. Furthermore, with the proposed NOMA-DRN with the JML, the full diversity order can be achieved in the low-medium SNR region.