Open-Set Source-Free Domain Adaptation aims to transfer knowledge in realistic scenarios where the target domain has additional unknown classes compared to the limited-access source domain. Due to the absence of information on unknown classes, existing methods mainly transfer knowledge of known classes while roughly grouping unknown classes as one, attenuating the knowledge transfer and generalization. In contrast, this paper advocates that exploring unknown classes can better identify known ones, and proposes a domain adaptation model to transfer knowledge on known and unknown classes jointly. Specifically, given a source pre-trained model, we first introduce an unknown diffuser that can determine whether classes in space need to be split and merged through similarity measures, to estimate and generate a wider class space distribution, including known and unknown classes. Based on such a wider space distribution, we enhance the reliability of known class knowledge in the source pre-trained model through contrastive constraint. Finally, various supervision information, including reliable known class knowledge and clustered pseudo-labels, optimize the model for impressive knowledge transfer and generalization. Extensive experiments show that our network can achieve superior exploration and knowledge generalization on unknown classes, while with excellent known class transfer. The code is available at https://github.com/xdwfl/UPUK.