Hierarchical open-set classification handles previously unseen classes by assigning them to the most appropriate high-level category in a class taxonomy. We extend this paradigm to the semi-supervised setting, enabling the use of large-scale, uncurated datasets containing a mixture of known and unknown classes to improve the hierarchical open-set performance. To this end, we propose a teacher-student framework based on pseudo-labeling. Two key components are introduced: 1) subtree pseudo-labels, which provide reliable supervision in the presence of unknown data, and 2) age-gating, a mechanism that mitigates overconfidence in pseudo-labels. Experiments show that our framework outperforms self-supervised pretraining followed by supervised adaptation, and even matches the fully supervised counterpart when using only 20 labeled samples per class on the iNaturalist19 benchmark. Our code is available at https://github.com/walline/semihoc.