In 2012 the authors set out a programme to prove the Duffin–Schaeffer conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a non-negative function ψ:N→Rψ:N→R, let W(ψ)W(ψ) denote the set of real numbers xx such that |nx−a|<ψ(n)|nx−a|<ψ(n) for infinitely many reduced rationals a/n (n>0)a/n (n>0). Our main result is that W(ψ)W(ψ) is of full Lebesgue measure if there exists a c>0c>0 such that ∑n≥16φ(n)ψ(n)nexp(c(loglogn)(logloglogn))=∞.