Summary: Here we prove the following result. Let \(X\) be a smooth and connected projective curve, \(x\) a non-negative integer, \(L\in\text{Pic}(X)\) and \(V\subseteq H^0(X,L)\) a linear subspace such that \(\dim (V)\geq 3x+4\), \(V\) spans \(L\) and the associated morphism \(\varphi_V: X\to\mathbb{P}^r\), \(f:=\dim(V)-1\), is unramified and birational onto its image. Then for general \(P_1,\dots,P_x\) the linear system \(V(-2P_1- \cdots-2P_x)\) has no base points, \(\dim(V(-2P_1- \cdots-2P_x))= \dim(V)-2x\) and the associated morphism \(\varphi_{V(-2P_1- \cdots-2P_x)}\) is unramified and birational onto its image.