In this article, we report a novel fiber-optic axial-strain sensor based on the bias-tapered micro-open-cavity (BT-MOC) structure. The comprehensive investigations of axial-strain response are theoretically and experimentally conducted with respect to the diameter and position of taper waist. The results show that the intensity variation is strongly related to the location of taper. When the diameter is ~ $30~\mu \text{m}$ , highly sensitive and linear intensity modulation (~0.043 dB/ $\mu \varepsilon)$ is gained by the bias position of ~ $210~\mu \text{m}$ , and for a small bias state, the wavelength response reaches 17 pm/ $\mu \varepsilon $ flatly and can be increased to 313.14 pm/ $\mu \varepsilon $ with the aid of Vernier effect. As a result, the detection resolution of ~ $0.2~\mu \varepsilon $ can be achieved in both intensity and wavelength modulated schemes. With the merits of compactness, repeatability, and low cost, our sensor is very promising and potential in the high-precision axial-strain-related engineering measurements.