Large language models (LLMs) enable zero-shot and few-shot multi-label text classification via in-context learning, yet most approaches perform static inference and degrade under streaming test data due to distribution shift and long-tail labels. We study online test-time adaptation for LLM-based multi-label generation without any parameter updates, and identify two bottlenecks: (1) standard generation probabilities provide unreliable confidence because they ignore label competition at key decoding branches; (2) naive confidence-based caching overfits to frequent and easy examples, reducing label coverage and diversity. We propose SCOTTA, a structured confidence-guided online adaptation framework. SCOTTA introduces Label-set Local Likelihood Ratio (L3R), a label-level confidence measure that compares a target label against its valid competitors at critical decision positions. Using L3R as a unified signal, SCOTTA maintains an in-context exemplar cache via streaming submodular maximization, balancing label coverage, semantic diversity, and sample quality under a fixed context budget. Across four benchmarks, SCOTTA consistently improves Micro-F1 and Macro-F1 over strong LLM and non-LLM baselines, with the largest gains on long-tail labels.