Integrating contact-awareness into a soft snake robot and efficiently controlling its locomotion in response to contact information present significant challenges. This paper aims to solve contact-aware locomotion problem of a soft snake robot through developing bio-inspired contact-aware locomotion controllers. To provide effective contact information for the controllers, we develop a scale covered sensor structure mimicking natural snakes' \textit{scale sensilla}. In the design of control framework, our core contribution is the development of a novel sensory feedback mechanism of the Matsuoka central pattern generator (CPG) network. This mechanism allows the Matsuoka CPG system to work like a "spine cord" in the whole contact-aware control scheme, which simultaneously takes the stimuli including tonic input signals from the "brain" (a goal-tracking locomotion controller) and sensory feedback signals from the "reflex arc" (the contact reactive controller), and generate rhythmic signals to effectively actuate the soft snake robot to slither through densely allocated obstacles. In the design of the "reflex arc", we develop two types of reactive controllers -- 1) a reinforcement learning (RL) sensor regulator that learns to manipulate the sensory feedback inputs of the CPG system, and 2) a local reflexive sensor-CPG network that directly connects sensor readings and the CPG's feedback inputs in a special topology. These two reactive controllers respectively facilitate two different contact-aware locomotion control schemes. The two control schemes are tested and evaluated in the soft snake robot, showing promising performance in the contact-aware locomotion tasks. The experimental results also further verify the benefit of Matsuoka CPG system in bio-inspired robot controller design.