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It is observed that previously proposed cooperative communication schemes do not increase throughput in the high signal-to-noise ratio (SNR) region although they improve the outage probability. In this paper we introduce a new cooperative communications protocol for direct-sequence code-division-multiple-access (DS-CDMA) systems that increases the SNR range over which there is no significant throughput loss compared to direct transmission. This eliminates the necessity for switching between more than one scheme which leads to higher system complexity. The proposed new scheme has a high spectral efficiency (reaching one asymptotically with the SNR and block size) at the cost of extra spreading sequences. However the channels are allowed to be non-orthogonal, and we analyze the effect of that on cooperative diversity under the LMMSE receiver. We present both achievable and outage throughput analysis under different cross correlations between user codes. From our results we show that the proposed scheme improves the throughput and the outage throughput performance in high SNR at the expense of only a negligible loss in the low SNR region compared to the previously proposed reference scheme.