To seek high signal-to-noise ratio (SNR) is critical but challenging for single-shot intense terahertz (THz) coherent detection. This paper presents an improved common-path spectral interferometer for single-shot THz detection with a single chirped pulse as the probe for THz electro-optic (EO) sampling. Here, the spectral interference occurs between the two orthogonal polarization components with a required relative time delay generated with only a birefringent plate after the EO sensor. Our experiments show that this interferometer can effectively suppress the noise usually suffered in a non-common-path interferometer. The measured single-shot SNR is up to 88.85, and the measured THz waveforms are independent of the orientation of the used ZnTe EO sensor, so it is easy to operate and the results are more reliable. These features mean that the interferometer is quite qualified for applications where strong THz pulses, usually with single-shot or low repetition rate, are indispensable.