Detecting microwave signals over a wide frequency range has numerous advantages as it enables simultaneous transmission of a large amount of information and access to more spectrum resources. This capability is crucial for applications such as microwave communication, remote sensing, and radar. However, conventional microwave receiving systems are limited by amplifiers and band-pass filters that can only operate efficiently in a specific frequency range. Typically, these systems can only process signals within a three-fold frequency range, which limits the data transfer bandwidth of the microwave communication systems. Developing novel atom-integrated microwave sensors, for example, radio frequency (RF)-chip coupled Rydberg atomic receiver, provides opportunities for a large working bandwidth of microwave sensing at the atomic level. Here, an ultra-wide dual-band RF sensing scheme is demonstrated by space-division multiplexing two RF-chip-integrated atomic receiver modules. The system can simultaneously receive dual-band microwave signals that span a frequency range exceeding 6 octaves (300 MHz and 24 GHz). This work paves the way for multi-band microwave reception applications within an ultra-wide range by RF-chip-integrated Rydberg atomic sensor.