We propose a new spread orthogonal frequency division multiplexing with index modulation (S-OFDM-IM), which employs precoding matrices such as Walsh-Hadamard (WH) and Zadoff-Chu (ZC) to spread both non-zero data symbols of active sub-carriers and their indices, and then compress them into all available sub-carriers. This aims to increase the transmit diversity, exploiting both multipath and index diversities. As for the performance analysis, we derive the bit error probability (BEP) to provide an insight into the diversity and coding gains, and especially impacts of selecting various spreading matrices on these gains. This interestingly reveals an opportunity of using rotated versions of original WH and ZC matrices to further improve the BEP performance. More specifically, rotated matrices can enable S-OFDM-IM to harvest the maximum diversity gain, which is the number of sub-carriers, while benchmark schemes have diversity gains limited by two. Moreover, we propose three low-complexity detectors, namely minimum mean square error log-likelihood ratio, index pattern MMSE (IP-MMSE), and enhanced IP-MMSE, which achieve different levels of complexity and reliability. Simulation results are presented to prove the superiority of S-OFDM-IM over the benchmarks.