We propose and demonstrate an extremely simple yet novel sensing strategy for measurements of a refractive index (RI) based on microwave-photonic optical fiber interferometry. A hybrid interferometric system based on an incoherent optical interferometer (i.e., a Michelson interferometer [MI]) and a coherent optical interferometer (i.e., a Fabry-Perot interferometer [FPI]) is constructed simply by using a low-cost off-the-shelf fiber coupler. The sensing arm of the MI is highly sensitive to a surrounding RI based on Fresnel reflection, where variations of the ambient RI cause changes in both the reflection magnitudes of the resonance frequencies and fringe visibility of the reflection spectra in the microwave domain. The coherent FPI is employed to tune the dynamic range of the MI by adjusting the effective reflectance of the reference arm of the MI. Essentially, other approaches that can vary the reflectance of the reference arm of the MI can also be used to tune the dynamic range of the system based on the proposed strategy. The experimental results are in good agreement with theoretical predictions. The prominent advantages of the sensor, including low cost, ease of fabrication, robustness, compactness, high sensitivity, and tunable dynamic range, make it a strong candidate in various chemical, biological, and environmental applications.