Summary: Let \(f\) be a holomorphic function of the unit disc \(\mathbb{D}\), preserving the origin. According to Schwarz’s Lemma, \(|f^{\prime}(0)|\leq1\), provided that \(f(\mathbb{D})\subset\mathbb{D}\). We prove that this bound still holds, assuming only that \(f(\mathbb{D})\) does not contain any closed rectilinear segment \([0,e^{i\phi}]\), \(\phi \in[0,2\pi]\), i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and we give a covering theorem.