Equipping a conversational search engine with strategies regarding when to ask clarification questions is becoming increasingly important across various domains. Attributing to the context understanding capability of LLMs and their access to domain-specific sources of knowledge, LLM-based clarification strategies feature rapid transfer to various domains in a post-hoc manner. However, they still struggle to deliver promising performance on unseen domains, struggling to achieve effective domain transferability. We take the first step to investigate this issue and existing methods tend to produce one-size-fits-all strategies across diverse domains, limiting their search effectiveness. In response, we introduce a novel method, called STYLE, to achieve effective domain transferability. Our experimental results indicate that STYLE bears strong domain transferability, resulting in an average search performance improvement of 10% on four unseen domains.