  This paper is a continuation of a previous work Germain-Pusateri (2020) 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
Germain-Pusateri (2020), but allow us to present some simplifications in the
proof of global existence with decay for small solutions. In particular, we can
propagate a stronger control on a basic $L^2$-weighted type norm while
providing some shorter and less technical proofs for some of the arguments.
