Large Language Models (LLMs) suffer from hallucination problems, which hinder their reliability in sensitive applications. In the black-box setting, several self-consistency-based techniques have been proposed for hallucination detection. We empirically study these techniques and show that they achieve performance close to that of a supervised (still black-box) oracle, suggesting little room for improvement within this paradigm. To address this limitation, we explore cross-model consistency checking between the target model and an additional verifier LLM. With this extra information, we observe improved oracle performance compared to purely self-consistency-based methods. We then propose a budget-friendly, two-stage detection algorithm that calls the verifier model only for a subset of cases. It dynamically switches between self-consistency and cross-consistency based on an uncertainty interval of the self-consistency classifier. We provide a geometric interpretation of consistency-based hallucination detection methods through the lens of kernel mean embeddings, offering deeper theoretical insights. Extensive experiments show that this approach maintains high detection performance while significantly reducing computational cost.