Cross-modal hashing that leverages hash functions to project high-dimensional data from different modalities into the compact common hamming space, has shown immeasurable potential in cross-modal retrieval. To ease labor costs, unsupervised cross-modal hashing methods are proposed. However, existing unsupervised methods still suffer from two factors in the optimization of hash functions: 1) similarity guidance, they barely give a clear definition of whether is similar or not between data points, leading to the residual of the redundant information; 2) optimization strategy, they ignore the fact that the similarity learning abilities of different hash functions are different, which makes the hash function of one modality weaker than the hash function of the other modality. To alleviate such limitations, this paper proposes an unsupervised cross-modal hashing method to train hash functions with discriminative similarity guidance and adaptively-enhanced optimization strategy, termed Deep Adaptively-Enhanced Hashing (DAEH). Specifically, to estimate the similarity relations with discriminability, Information Mixed Similarity Estimation (IMSE) is designed by integrating information from distance distributions and the similarity ratio. Moreover, Adaptive Teacher Guided Enhancement (ATGE) optimization strategy is also designed, which employs information theory to discover the weaker hash function and utilizes an extra teacher network to enhance it. Extensive experiments on three benchmark datasets demonstrate the superiority of the proposed DAEH against the state-of-the-arts.