Single-shot recording of terahertz electric signals has recently become possible at high repetition rates, by using the photonic time-stretch electro-optic sampling (EOS) technique. However the moderate sensitivity of time-stretch EOS is still a strong limit for a range of applications. Here we present a variant enabling to increase the sensitivity of photonic time-stretch for free-propagating THz signals. The ellipticity of the laser probe is enhanced by adding a set of Brewster plates, as proposed by Ahmed et al. [Rev. Sci. Instrum. 85, 013114 (2014)] in a different context. The method is tested using the high repetition rate terahertz coherent synchrotron radiation source of the SOLEIL synchrotron radiation facility. The signal-to-noise ratio of our terahertz digitizer could thus be straightforwardly improved by a factor ≈6.5, leading to a noise-equivalent input electric field below 1.25 V/cm inside the electro-optic crystal, over the 0-300 GHz band (i.e., 2.3 μV/cm/Hz). The sensitivity is scalable with respect to the available laser power, potentially enabling further sensitivity improvements when needed.