What is believed to be a novel long-period grating (LPG) refractive index sensor with a modified cladding structure is proposed and studied. In the proposed structure, the cladding of the fiber has a two-layer structure, i.e., a cladding layer of low refractive index with a reduced radius and an overlay of high refractive index. The sensitivity of the structure-modified LPG sensor to the ambient refractive index change as a function of the cladding layer and overlay parameters is investigated by way of modeling. It is found that an increase of the ambient refractive index causes a field redistribution of the cladding mode into the overlay when the parameters of the overlay are properly selected. It is shown that by reducing the radius of the cladding layer, the operational range of the LPG refractive index sensor can be as large as 0.195 (from 1.244 to 1.440) with a minimum sensitivity of 660 nm/refractive index, which represents a 31% increase of operational range in comparison with the operational range obtained from the reported structure. The design guidelines for achieving this large operation range and high sensitivity are explained by investigating the dependence of the cladding modes on the radius of the cladding layer.