This paper proposes a new framework of real-coded genetic algorithms (RCGAs) for the multi-funnel function optimization. The RCGA is one of the most powerful function optimization methods. Most conventional RCGAs work effectively on the single-funnel function that consists of a single big-valley. However, it is reported that they show poor performance or, sometimes, fail to find the optimum on the multi-funnel function that consists of multiple big-valleys. In order to remedy this deterioration, Innately Split Model (ISM) has been proposed as a framework of RCGAs. ISM initializes an RCGA in a small region and repeats a search with the RCGA as changing the position of the region randomly. ISM outperforms conventional RCGAs on the multi-funnel functions. However, ISM has two problems in terms of the search efficiency and the difficulty of setting parameters. Our proposed method, Big-valley Explorer (BE), is a framework of RCGAs like ISM and it has two novel mechanisms to overcome these problems, the big-valley estimation mechanism and the adaptive initialization mechanism. Once the RCGA finishes a search, the big-valley estimation mechanism estimates a big-valley that the RCGA already explored and removes the region from the search space to prevent the RCGA from searching the same big-valley many times. After that, the adaptive initialization mechanism initializes the RCGA in a wide unexplored region adaptively to find unexplored big-valleys. We evaluate BE through some numerical experiments with both single-funnel and multi-funnel benchmark functions.