Graphene has been known to possess exceptional mechanical properties, including its extremely high Young's modulus and atomic layer thickness. Although there are several reported fiber optic pressure sensors using graphene film, a key question that is not well understood is how the suspended graphene film interacts with the backing air cavity and affects the sensor performance. Based on our previous analytical model, we will show that the sensor performance suffers due to the significantly reduced mechanical sensitivity by the backing cavity. To remedy this limitation, we will, through experimental and numerical methods, investigate two approaches to enhance the sensitivity of fiber optic acoustic pressure sensors using graphene film. First, a graphene-silver composite diaphragm is used to enhance the optical sensitivity by increasing the reflectivity. Compared with a sensor with pure graphene diaphragm, graphene-silver composite can enhance the sensitivity by threefold, while the mechanical sensitivity is largely unchanged. Second, a fiber optic sensor is developed with enlarged backing air volume through the gap between an optical fiber and a silica capillary tube. Experimental results show that the mechanical sensitivity is increased by 10× from the case where the gap side space is filled. For both approaches, signal-to-noise ratio (SNR) is improved due to the enhanced sensitivity, and COMSOL Thermoviscous acoustics simulation compares well with the experimental results. This study is expected to not only enhance the understanding of fluid-structural interaction in sensor design but also benefit various applications requiring high-performance miniature acoustic sensors.