Let \(00\), \(i=1,\,2,\,\cdots,\,m-2\), be given constants with \(\sum_{i=1}^{m-2}a_i=1\). This paper studies the existence of a solution to the second-order multipoint boundary value problem \[ x''=f(t,\,x,\,x')=0,\qquad t\in (0,\,1),\quad x'(0)=0,\;x(1)=\sum_{i=1}^{m-2}a_i u(\xi_i), \] where \(f:[0,\,1]\times \mathbb{R}^2\to \mathbb{R}\) is continuous. The author improves some results of his paper [J. Math. Anal. Appl. 211, 545--555 (1997; Zbl 0884.34024)], where the essential condition, \(x\,f(t,\,x,\,0)\geq \delta\) for \(| x| >M\), where \(M\) and \(\delta\) are positive constants, is replaced by the weaker sign conditions that \(f(t,\,-M,\,0)\leq 0\) and \(f(t,\,M,\,0)\geq 0\). The arguments are based on the nonlinear alternative of Leray-Schauder.