We have shown that strongly overcoupled resonant modes of long-period fiber gratings (LPFGs) can be used to assess the adulteration of olive oil. In this background, we investigate the response characteristics of strongly overcoupled resonant modes of different orders to a surrounding medium, the refractive index (RI) of which is greater than that of the fiber cladding, and in the range where a precise refractive index measurement is immensely useful for inspecting the quality of olive oils and other edible oils. A theoretical simulation that would help in designing a sensor with suitable sensitivity and range of measurement has been presented in detail and also validated with experimental results. It was interesting to observe that in a high RI surrounding, a lower order overcoupled resonant mode is much more sensitive as compared to a higher-order one having a similar coupling coefficient. A quantitative analysis demonstrates that for a particular LPFG, the sensitivity of a strongly overcoupled LP06 mode was found to be ∼2000 dB/RIU, while that of the LP07 mode having similar coupling strength was ∼550 dB/RIU in the surrounding refractive index range from 1.458 to 1.520. The results have been validated experimentally.