We consider the Stein exponent of distributed hypothesis testing (in the side-information setting). Decades since the problem was first formulated, the exponent is still an open problem, except for some special cases. Rahman and Wagner have derived an upper bound, by providing the decoder with side information that creates conditional independence, where single-letterization is possible. We propose a new technique, inspired by their work, which provides side information in a more gradual manner. For the special case of testing for Gaussian correlations, we show that our technique strictly improves upon the known bounds, and in particular it gives a finite upper bound for parameters where no such bound existed.