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The recorded spectral fringes are usually evaluated by Fourier-transform (FT) based methods, that require high spectral resolution. Considering that most spectrometers in the mid-infrared (MIR) region have typically a few nm spectral resolution, we propose two alternative methods for measurements in this spectral region: second harmonic assisted SRI (SH-SRI) and dual-band SRI. The SH-SRI method utilizes second harmonic generation, which allows for high-precision phase retrieval by shifting the detection range from the MIR regime to the near-infrared (NIR), where high-resolution spectrometers are commercially available. The dual-band SRI method has the capabilities of the SH-SRI, while simultaneously probing the optical element at the SH frequency band as well. Using the FT method, the acquired phase can be obtained from a single interferogram at both frequency bands simultaneously, which effectively doubles the measurement bandwidth compared to the conventional and the SH-SRI without increasing the requirements on the bandwidth and the resolution of the spectrometer. To determine the performance of these new schemes, we measured the spectral phase shift of several optical windows, and concluded that SH-SRI and dual-band SRI both have higher accuracy than conventional SRI using low-resolution MIR spectrometers. We used the SH-SRI technique also to verify the phase performance of custom-made MIR dispersive mirrors. 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