Social determinants of health (SDOH) are nonclinical, socioeconomic conditions that influence patient health and quality of life. Identifying SDOH may help clinicians target interventions. However, SDOH are more frequently available in narrative notes compared to structured electronic health records. The 2022 n2c2 Track 2 competition released clinical notes annotated for SDOH to promote development of NLP systems for extracting SDOH. We developed a system addressing 3 limitations in state-of-the-art SDOH extraction: the inability to identify multiple SDOH events of the same type per sentence, overlapping SDOH attributes within text spans, and SDOH spanning multiple sentences. We developed and evaluated a 2-stage architecture. In stage 1, we trained a BioClinical-BERT-based named entity recognition system to extract SDOH event triggers, that is, text spans indicating substance use, employment, or living status. In stage 2, we trained a multitask, multilabel NER to extract arguments (eg, alcohol "type") for events extracted in stage 1. Evaluation was performed across 3 subtasks differing by provenance of training and validation data using precision, recall, and F1 scores. When trained and validated on data from the same site, we achieved 0.87 precision, 0.89 recall, and 0.88 F1. Across all subtasks, we ranked between second and fourth place in the competition and always within 0.02 F1 from first. Our 2-stage, deep-learning-based NLP system effectively extracted SDOH events from clinical notes. This was achieved with a novel classification framework that leveraged simpler architectures compared to state-of-the-art systems. Improved SDOH extraction may help clinicians improve health outcomes.