In this paper, we propose techniques that reduce the MAC overhead in IEEE 802.11-based wireless networks. As the channel bit rate increases, MAC overhead becomes a major source of throughput degradation. This is due to the fact that the MAC overhead is rate-independent; it is fixed regardless of the data rate. Rate-independent overhead can be reduced by dividing a high-rate wideband channel into multiple low-rate narrow channels, and send data on these narrow channels simultaneously using multiple or compound radios. In this environment with multiple narrow channels, we propose two techniques that further reduce MAC overhead and thus improve resource efficiency. First, channel contention is done only on one of the narrow channels, called the contention channel. The other channels are accessed based on the status of the contention channel, without contention. This technique significantly improves network throughput, but has a problem in which if a collision occurs on the contention channel, collisions can also occur on other channels. To avoid this phenomenon, we propose the second technique in which a node observing collision notifies the senders that cause collision, so that they do not access other channels simultaneously. Extensive simulation results show that the proposed techniques improve network throughput by a considerable amount.