In this article, we demonstrate an improved efficient fibre sensor with a high sensitivity to measure glucose concentrations in the physiological range of human beings, operating in a broad spectral bandwidth from the near- to mid-infrared. The sensor consists of a dual-peak long period grating (DPLPG) with a period of 150 ÎŒm inscribed in an optical fibre with a diameter of 80 ÎŒm. The investigation of sensing for refractive index results in a sensitivity of ~-885.7 nm/refractive index unit (RIU) and ~2008.6 nm/RIU in the range of 1.30-1.44. The glucose measurement is achieved by the immobilisation of a layer of enzyme of glucose oxidase (GOD) onto the fibre surface for the selective enhancement of sensitivity for glucose. The sensor can measure glucose concentrations with a maximum sensitivity of -36.25 nm/(mg/mL) in the range of 0.1-3.0 mg/mL. To the best of our knowledge, this is the highest sensitivity ever achieved for a measurement of glucose with a long period grating-based sensor, indicating its potential for many applications including pharmaceutical, biomedical and food industries.