Recent years have witnessed the great success of hyper-heuristics applying to numerous real-world applications. Hyper-heuristics raise the generality of search methodologies by manipulating a set of low level heuristics (LLHs) to solve problems, and aim to automate the algorithm design process. However, those LLHs are usually parameterized, which may contradict the domain independent motivation of hyper-heuristics. In this paper, we show how to automatically maintain low level parameters (LLPs) using a hyper-heuristic with LLP adaptation (AD-HH), and exemplify the feasibility of AD-HH by adaptively maintaining the LLPs for two hyper-heuristic models. Furthermore, aiming at tackling the search space expansion due to the LLP adaptation, we apply a heuristic space reduction (SAR) mechanism to improve the AD-HH framework. The integration of the LLP adaptation and the SAR mechanism is able to explore the heuristic space more effectively and efficiently. To evaluate the performance of the proposed algorithms, we choose the p-median problem as a case study. The empirical results show that with the adaptation of the LLPs and the SAR mechanism, the proposed algorithms are able to achieve competitive results over the three heterogeneous classes of benchmark instances.