Deep unfolding networks (DUNs) have attracted growing attention in compressed sensing (CS) due to their good interpretability and high performance. However, many DUNs often improve the reconstruction effect at the price of a large number of parameters and have the problem of feature information loss during iteration. This paper proposes a novel adaptive memory-augmented unfolding network for compressed sensing (AMAUN-CS). Concretely, without loss of interpretability, we integrate an adaptive content-aware strategy into the gradient descent step of the proximal gradient descent (PGD) algorithm, driving it to adaptively capture the adequate features. In addition, we extended AMAUN-CS based on the memory storage mechanism of the human brain and propose AMAUN-CS+ to develop the dependency of deep information across cascading stages. The experimental results show that the AMAUN-CS model surpasses other advanced methods on various public benchmark datasets while having lower complexity in training.