This paper firstly and experimentally demonstrates an in-fiber axial micro-strain sensing head, combined with a Mach-Zehnder interferometer (MZI) based on the concentric multilayer elliptical-core fiber (CMECF). This MZI with a high extinction ratio (about 15 dB) is successfully achieved with a CMECF-single mode fiber-CMECF (CSC) structure. The MZI sensor theory and the resonance demodulation technology are systematically described in this paper. In this CSC structure, two sections of the CMECF have a role as the mode generator and coupler, respectively. LP01 and LP11 even, which have similar excitation coefficients, are two dominated propagating mode groups supported in the CMECF. On account of the distinct dual-mode property, a good stability of this sensor is realized. The detected resonance in the MZI shifts as the axial micro-strain variated due to the strong interaction between higher order modes. High sensitivity of ~1.78 pm/με is experimentally achieved within the range of 0 με - 1250 με, meanwhile, the intensity fluctuation is below 0.38 dB.