To break through the limitations of existing hash functions based on quantum walks with memory in diffusion efficiency and step-length control, we develop a memory-driven controlled alternate quantum walks model and propose a corresponding hash function. Firstly, the proposed model integrates lively quantum walks with quantum walks with memory and introduces a two-step memory decision mechanism. This mechanism allows historical memory to dynamically regulate the jumping amplitude in the active direction, thereby strengthening the coupling between memory and the coin operator and accelerating the diffusion of the path. Secondly, on the basis of this model, we propose a quantum hash function with variable-length output. Experimental evaluations demonstrate strong input sensitivity, robust collision resistance, and effective output confusion. Moreover, the output distribution is balanced, the perturbation response is highly sensitive, and a pronounced avalanche effect is consistently observed. Our work introduces a novel path control paradigm for the systematic design of quantum hash functions built on quantum walks.