User relaying in cooperative non-orthogonal multiple access (C-NOMA) system is able to enhance reliability. However, for user operates in half-duplex (HD) mode, additional bandwidth is needed for relaying operation. This reduces spectral efficiency of C-NOMA system. This letter proposes a successive user relaying (SR) to improve the spectral efficiency of C-NOMA system. In the proposed SR scheme, a pair of HD users decode the received signal and relay the decoded signal successively to improve the spectral efficiency. A novel algorithm is proposed to optimize power split factor in order to improve fairness of outage performance between users. The performance of the proposed SR scheme is superior to existing HD relaying scheme in C-NOMA system and existing SR in cooperative orthogonal multiple access system in terms of outage probability and ergodic sum rate. The closed-form expressions of outage probability and asymptotic ergodic sum rate of proposed SR are also derived.