We develop a versatile, compact, and sensitive fiber-based optical Fabry-Pérot (FP) gas sensor. The sensor probe is composed of a silver layer and a vapor-sensitive polymer layer that are sequentially deposited on the cleaved fiber endface, thus forming an FP cavity. The interference spectrum resulting from the reflected light at the silver-polymer and polymer-air interfaces changes when the polymer is exposed to gas analytes. This structure enables using any polymer regardless of the polymer refractive index (RI), which significantly enhances the sensor versatility. In experiments, we use polyethylene glycol (PEG) 400 (RI=1.465-1.469) and Norland Optical Adhesive (NOA) 81 (RI=1.53-1.56) as the gas sensing polymer and show drastically different sensor response to hexanol, methanol, and acetone. The estimated sensitivity for methanol vapor is 3.5 pm/ppm and 0.1 pm/ppm for PEG 400 and NOA 81, respectively, with a detection limit on the order of 1-10 ppm. Gas sensing for the analytes delivered in both continuous flow mode and pulsed mode is demonstrated.