This paper is a continuation of the previous work (Pusateri, in: Forum of mathematics, Cambridge University Press) by the first two authors. We focus on 1-dimensional quadratic Klein–Gordon equations with a potential, under some assumptions that are less general than (Pusateri, in: Forum of mathematics, Cambridge University Press), but that allow us to present some simplifications in the proof of the global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic L2-weighted-type norm while providing some shorter and less technical proofs for some of the arguments.