Understanding dataset semantics is crucial for effective search, discovery, and integration pipelines. To this end, column type annotation (CTA) methods associate columns of tabular datasets with semantic types that accurately describe their contents, using pre-trained deep learning models or Large Language Models (LLMs). However, existing approaches require users to specify a closed set of semantic types either at training or inference time, hindering their application to domain-specific datasets where pre-defined labels often lack adequate coverage and specificity. Furthermore, real-world datasets frequently contain columns with values belonging to multiple semantic types, violating the single-type assumption of existing CTA methods. While proprietary LLMs have shown effectiveness for CTA, they incur high monetary costs and produce inconsistent outputs for similar columns, leading to type redundancy that negatively affects downstream applications. To address these challenges, we introduce StraTyper, a cost-effective method for column type discovery (CTD) and multi-type annotation (CMTA) in dataset collections. StraTyper eliminates the need for pre-defined semantic labels by systematically employing LLMs to discovery types tailored to the dataset collection at hand. Through strategic column clustering, controlled type generation, and iterative cascading discovery, StraTyper balances type precision with annotation coverage while minimizing LLM costs. Our experimental evaluation-both manual and LLM-assisted-on real-world benchmarks demonstrates that StraTyper discovers accurate types for both numerical and non-numerical data, achieves substantial cost savings compared to commercial LLMs, and effectively handles multi-typed columns. We further show that StraTyper's annotations improve downstream tasks, including join discovery and schema matching, outperforming LLM-only baselines.