Chatbots' role in fostering self-reflection is now widely recognized, especially in inducing users' behavior change. While the benefits of 24/7 availability, scalability, and consistent responses have been demonstrated in contexts such as healthcare and tutoring to help one form a new habit, their utilization in coaching necessitating deeper introspective dialogue to induce leadership growth remains unexplored. This paper explores the potential of such a chatbot powered by recent Large Language Models (LLMs) in collaboration with professional coaches in the field of executive coaching. Through a design workshop with them and two weeks of user study involving ten coach-client pairs, we explored the feasibility and nuances of integrating chatbots to complement human coaches. Our findings highlight the benefits of chatbots' ubiquity and reasoning capabilities enabled by LLMs while identifying their limitations and design necessities for effective collaboration between human coaches and chatbots. By doing so, this work contributes to the foundation for augmenting one's self-reflective process with prevalent conversational agents through the human-in-the-loop approach.