In this paper, we propose a new AF cooperative transmission scheme with multiple relays applying adaptive modulation to maximize the system spectral efficiency. We quantify the average spectral efficiency performance of the proposed scheme through statistical analysis based on an upper bound of the combined output SNR and obtain the closed-form solutions for some practical scenarios of interest. Selected numerical results demonstrate that the analytical results have a tight upper bound and our proposed scheme leads to a considerable spectral efficiency gain in the cooperative diversity systems.