Dual mode orthogonal frequency division multiplexing with index modulation (DM-OFDM-IM) is recently proposed, which modulates all subcarriers to eliminate the limits of spectrum efficiency in OFDM with index modulation (OFDM-IM). In DM-OFDM-IM, the subcarriers within each subblock are divided into two groups, which are modulated by two distinguishable constellation alphabets drawn from the distinct two constellations. In this paper, a new constellation design for DM-OFDM-IM is proposed based on the pairwise error probability (PEP) analysis. In previous works, the design of the constellation pair is naive and induces much power. Also, a new bit mapping scheme for DM-OFDM-IM is proposed. The proposed bit mapping scheme can relieve the catastrophic effect of the index demodulation error incurring burst of errors, which is the endemic problem of the index demodulation systems. The simulation results show that the proposed constellation pair and the proposed bit mapping scheme substantially enhance the BER performance of the DM-OFDM-IM than the conventional DM-OFDM-IM.