Open-set domain adaptation (OSDA) transfers knowledge to an unlabeled target domain under both distribution shift and unknown classes absent in the source domain. Most OSDA methods require access to both source and target data and rely on either feature-space or logit-space information for known-unknown separation. However, source data is often restricted due to storage or privacy constraints, and single-space reliance can weaken separation, as unknown samples may be distinguishable in one space but not the other. To address these limitations, we propose Progressive Dual-Space Discovering (PDD), a source-free OSDA method that progressively adapts a pre-trained model for improved domain alignment and known-unknown separation. PDD iteratively builds a credible domain by selecting target samples close to the known-class distribution through dual-space selection: energy-based filtering in logit space followed by prototype-based refinement in feature space. Besides, PDD performs clustering using feature-space information from the credible domain and logit-space information from previously trained models, forming known and unknown domains. With these established domains, cross-entropy loss optimizes learning within the credible domain, while HSIC loss aligns the credible and known domains. Additionally, dual-space uncertainty losses enhance the separation between known and unknown classes. Extensive experiments on three OSDA benchmarks demonstrate the effectiveness of dual-space discovering, known-unknown separation, and progressive updates, facilitating PDD to achieve state-of-the-art performance. Code is available at https://github.com/qszhan/PDD .