To overcome the inherently low temperature sensitivity of fiber Bragg gratings (FBGs) in engineering applications under low-temperature conditions, a sensitivity-enhanced FBG temperature sensor based on a polytetrafluoroethylene (PTFE) encapsulation sleeve was developed. Four adhesive materials—silicone thermal grease, polydimethylsiloxane (PDMS), epoxy resin, and modified acrylic ester—were employed to package the FBG within the PTFE sleeve to improve its temperature sensitivity. Thermal stress simulations of the proposed sensor structure were carried out using COMSOL Multiphysics^® 6.2, and the simulation results showed good agreement with the experimental data. Based on the experimental results, the sensitivity-enhancement effects of PTFE combined with different adhesives, as well as the influences of the PTFE sleeve length and wall thickness, were systematically investigated. The results indicate that, within the temperature range of −35 °C to 15 °C, increasing both the length and thickness of the PTFE sleeve can effectively improve the temperature sensitivity of the sensor. When epoxy resin was used as the encapsulating adhesive, the sensor achieved a maximum sensitivity of 117.4 pm/°C, corresponding to a 13.19-fold increase compared with that of a bare FBG sensor. This sensitivity-enhancing packaging structure significantly improves both the temperature sensitivity and linearity of FBG temperature sensors, while also substantially reducing fabrication costs.