Previous research on persona-based dialogue agents typically preset the agent's persona before deployment, which remains static thereafter. In this paper, we take a step further and explore a new paradigm called Self-evolving Personalized Dialogue Agents (SPDA), where the agent continuously evolves during the conversation to better align with the user's anticipation by dynamically adapting its persona. This paradigm could enable better personalization for each user, but also introduce unique challenges, which mainly lie in the process of persona adaptation. Two key issues include how to achieve persona alignment with the user and how to ensure smooth transition in the adaptation process. To address them, we propose a novel framework that refines the persona at hierarchical levels to progressively align better with the user in a controllable way. Experiments show that integrating the personas adapted by our framework consistently enhances personalization and overall dialogue performance across various base systems.