Most sensors face a common trade-off between high sensitivity and a large dynamic range. We demonstrate here an all-fiber refractometer based on a dual-cavity Fabry-Perot interferometer (FPI) that possesses the advantage of both high sensitivity and a large dynamic range. Since the two composite cavities have a large cavity length difference, one can observe both fine and coarse fringes, which correspond to the long cavity and the short cavity, respectively. The short-cavity FPI and the use of an intensity demodulation method mean that the individual fine fringe dips correspond to a series of quasi-continuous highly sensitive zones for refractive index measurement. By calculating the parameters of the composite FPI, we find that the range of the ultra-sensitive zones can be considerably adjusted to suit the end requirements. The experimental trends are in good agreement with the theoretical predictions. The co-existence of high sensitivity and a large dynamic range in a composite FPI is of great significance to practical RI measurements.