The effectiveness of search systems is evaluated using relevance labels that indicate the usefulness of documents for specific queries and users. While obtaining these relevance labels from real users is ideal, scaling such data collection is challenging. Consequently, third-party annotators are employed, but their inconsistent accuracy demands costly auditing, training, and monitoring. We propose an LLM-based modular classification pipeline that divides the relevance assessment task into multiple stages, each utilising different prompts and models of varying sizes and capabilities. Applied to TREC Deep Learning (TREC-DL), one of our approaches showed an 18.4% Krippendorff's $α$ accuracy increase over OpenAI's GPT-4o mini while maintaining a cost of about 0.2 USD per million input tokens, offering a more efficient and scalable solution for relevance assessment. This approach beats the baseline performance of GPT-4o (5 USD). With a pipeline approach, even the accuracy of the GPT-4o flagship model, measured in $α$, could be improved by 9.7%.