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Sparse signal recovery problems from noisy linear measurements appear in many areas of wireless communications. In recent years, deep learning (DL) based approaches have attracted interests of researchers to solve the sparse linear inverse problem by unfolding iterative algorithms as neural networks. Typically, research concerning DL assume a fixed number of network layers. However, it ignores a key character in traditional iterative algorithms, where the number of iterations required for convergence changes with varying sparsity levels. By investigating on the projected gradient descent, we unveil the drawbacks of the existing DL methods with fixed depth. Then we propose an end-to-end trainable DL architecture, which involves an extra halting score at each layer. Therefore, the proposed method learns how many layers to execute to emit an output, and the network depth is dynamically adjusted for each task in the inference phase. We conduct experiments using both synthetic data and applications including random access in massive MTC and massive MIMO channel estimation, and the results demonstrate the improved efficiency for the proposed approach.