In this paper, based on the architecture of diversity-achieving quadrature spatial modulation (DA-QSM) transmission scheme, we propose an Enhanced DA-QSM (EDA-QSM) scheme. In the proposed EDA-QSM scheme, by using Kronecker product extension to the eight dispersion matrices of Sezginer-Sari-Biglieri (SSB) code, the size of the dispersion matrices set is doubled compared with DA-QSM. Consequently, EDA-QSM is shown to have higher spectral efficiency as well as nonvanishing determinant (NVD) property without requiring any parameter optimization. Furthermore, EDA-QSM with an appropriate ordering of the dispersion matrices is a block-orthogonal space-time code (BOSTC), which then makes the four real variables corresponding to the same block are independent and orthogonal. By utilizing this feature, for the detection of EDA-QSM we propose a state codewords matched block-by-block sphere decoding (SD) algorithm, which significantly decreases the decoding complexity of ML decoder. Simulation results of bit error rate (BER) performance show that the new EDA-QSM scheme outperforms some newly presented diversity-achieving SM schemes.