Open-set semi-supervised object detection (OSSOD) methods aim to utilize practical unlabeled datasets with out-of-distribution (OOD) instances for object detection. The main challenge in OSSOD is distinguishing and filtering the OOD instances from the in-distribution (ID) instances during pseudo-labeling. The previous method uses an offline OOD detection network trained only with labeled data for solving this problem. However, the scarcity of available data limits the potential for improvement. Meanwhile, training separately leads to low efficiency. To alleviate the above issues, this paper proposes a novel end-to-end online framework that improves performance and efficiency by mining more valuable instances from unlabeled data. Specifically, we first propose a semi-supervised OOD detection strategy to mine valuable ID and OOD instances in unlabeled datasets for training. Then, we constitute an online end-to-end trainable OSSOD framework by integrating the OOD detection head into the object detector, making it jointly trainable with the original detection task. Our experimental results show that our method works well on several benchmarks, including the partially labeled COCO dataset with open-set classes and the fully labeled COCO dataset with the additional large-scale open-set unlabeled dataset, OpenImages. Compared with previous OSSOD methods, our approach achieves the best performance on COCO with OpenImages by +0.94 mAP, reaching 44.07 mAP.