Reactive navigation is a well-known paradigm for controlling an autonomous mobile robot, which suggests making all control decisions through some light processing of the current/recent sensor data. Among the many advantages of this paradigm are: 1) the possibility to apply it to robots with limited and low-priced hardware resources, and 2) the fact of being able to safely navigate a robot in completely unknown environments containing unpredictable moving obstacles. As a major disadvantage, nevertheless, the reactive paradigm may occasionally cause robots to get trapped in certain areas of the environment—typically, these conflicting areas have a large concave shape and/or are full of closely-spaced obstacles. In this last respect, an enormous effort has been devoted to overcome such a serious drawback during the last two decades. As a result of this effort, a substantial number of new approaches for reactive navigation have been put forward. Some of these approaches have clearly improved the way how a reactively-controlled robot can move among densely cluttered obstacles; some other approaches have essentially focused on increasing the variety of obstacle shapes and sizes that could be successfully circumnavigated; etc. In this paper, as a starting point, we choose the best existing reactive approach to move in densely cluttered environments, and we also choose the existing reactive approach with the greatest ability to circumvent large intricate-shaped obstacles. Then, we combine these two approaches in a way that makes the most of them. From the experimental point of view, we use both simulated and real scenarios of challenging complexity for testing purposes. In such scenarios, we demonstrate that the combined approach herein proposed clearly outperforms the two individual approaches on which it is built.