In this article we consider a lattice system of unbounded continuos spins. Otto & Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions. For the proof a more detailed analysis based on two new ingredients is needed. The two new ingredients are a new basic covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations for the conditioned Gibbs measures are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. The latter simplifies the application of the main result. As an application, we show how decay of correlations combined with the uniform LSI yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida form finite-range to infinite-range interaction.