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IEEE 802.11 MAC, called the Distributed Coordination Function (DCF), employs carrier sensing to effectively avoid collisions, but this makes it difficult to maximally reuse the spatial spectral resource available for exposed terminals. This paper proposes a new MAC algorithm, called Multiple Access with Salvation Army (MASA), which adopts less sensitive carrier sensing to promote more spatial reuse of the channel. However, this may result in a higher collision probability. MASA alleviates this problem by adaptively adjusting the communication distance via "packet salvaging" at the MAC layer. Extensive simulation based on the ns-2 has shown MASA to offer as much as 25% higher packet delivery rate and 27% higher throughput than the DCF with the CBR (constant bit rate) and TCP traffic, respectively. In particular, a significant reduction in packet delay, 86% and 70% lower packet delay with the CBR and TCP traffic, makes MASA suitable for delay-sensitive applications. For practicality, we discuss the implementation of MASA based on the DCF specification.