The composed image retrieval (CIR) task aims to retrieve the desired target image for a given multimodal query, i.e., a reference image with its corresponding modification text. The key limitations encountered by existing efforts are two aspects: 1) ignoring the multiple query-target matching factors; 2) ignoring the potential unlabeled reference-target image pairs in existing benchmark datasets. To address these two limitations is non-trivial due to the following challenges: 1) how to effectively model the multiple matching factors in a latent way without direct supervision signals; 2) how to fully utilize the potential unlabeled reference-target image pairs to improve the generalization ability of the CIR model. To address these challenges, in this work, we first propose a CLIP-Transformer based muLtIfactor Matching Network (LIMN), which consists of three key modules: disentanglement-based latent factor tokens mining, dual aggregation-based matching token learning, and dual query-target matching modeling. Thereafter, we design an iterative dual self-training paradigm to further enhance the performance of LIMN by fully utilizing the potential unlabeled reference-target image pairs in a weakly-supervised manner. Specifically, we denote the iterative dual self-training paradigm enhanced LIMN as LIMN+. Extensive experiments on four datasets, including FashionIQ, Shoes, CIRR, and Fashion200 K, show that our proposed LIMN and LIMN+ significantly surpass the state-of-the-art baselines.