Summary: This note sharpens the smoothing inequality of G. Giacomin and F. L. Toninelli [Commun. Math. Phys. 266, No. 1, 1–16 (2006; Zbl 1113.82032); “Smoothing of depinning transitions for directed polymers with quenched disorder”, Phys. Rev. Lett. 96, No. 7, Article ID 070602 (2006; doi:10.1103/PhysRevLett.96.070602] for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.