Offset spatial modulation orthogonal frequency division multiplexing (OSM-OFDM) is demonstrated to offer a simple radio frequency (RF) switching structure with comparable bit error rate (BER) performance to conventional SM-OFDM. However, the OSM-OFDM waveform inherits the large peakto-average power ratio (PAPR) of original OFDM, and suffers performance loss in terms of BER due to the nonlinear noise from high power amplifier (HPA). As a class of efficient PAPR reduction techniques, the BER performance of clipping-aided OSM-OFDM is first evaluated through theoretical analysis. Furthermore, in this contribution, we conceive a new antenna offset structure as grouped OSM-OFDM toward low PAPR, while its upper BER bound is quantified through theoretical analysis. Through simulation results, we demonstrate that compared to the original OSM-OFDM, both grouped OSM-OFDM and clipped OSM-OFDM are capable of striking a better balance between PAPR and BER performance, while the former one is without extra computational complexity and the later one needs distortion recovery at the receiver side.