Luby transform (LT) codes provide an efficient way to transfer information over erasure channels. Past research has shown that LT codes can perform well for a large number of input symbols. However, mathematical analysis and simulation results have revealed that the packet overhead for LT decoders can be as large as 100% when the number of input symbols is small. Designing an efficient decoder to handle a small number of symbols becomes an imminent research issue. In this paper, we make an observation that LT decoders often fail to recover all the input symbols, while LT encoders have a high probability of producing a full-rank coefficient matrix. Motivated by this observation, we propose a novel decoding algorithm called LT-W, in which we incorporate the use of the Wiedemann solver into LT decoding to extend the decodability of LT codes. Extensive experiments show that our proposed method reduces the packet overhead significantly and yet preserves the efficiency of the original LT decoding process.