We propose a near intercarrier interference (ICI)-free and very low complexity iterative detector for frequency-asynchronous distributed Alamouti-coded (FADAC) orthogonal frequency division multiplexing (OFDM). In the previous cancelation schemes, the entire subcarrier signals from one transmit (TX) antenna are estimated and canceled in the received signal from the other TX antenna and vice versa. However, the reliability of the estimated symbols are revealed to significantly vary across the subcarriers and thus, the poorly estimated symbols lead to the incorrect cancelation. Motivated from this, we first propose a scheme which does not cancel the interfering subcarrier(s) at the half band edges which undergo very high interference in FADAC-OFDM. For further improvement, we propose a so-called selective scheme which instantly measures the reliability of the detected symbols at each iteration and then exclude the unreliable symbols in the estimated interference generation. Moreover, the proposed scheme has a drastically reduced complexity by converting the cancelation process from the subcarrier domain to the time domain. In accordance with the analysis on the considered reliability measures, the numerical results show that the proposed scheme achieves the near ICI-free level only within three or four iterations for wide ranges of SNR, frequency offset, and delay spread.