We study the stochastic recursion $X_n=Ψ_n(X_{n-1})$, where $(Ψ_n)_{n\geq 1}$ is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation $x\mapsto Ax+B$. We describe the tail behaviour of the stationary solution $X$ under the assumption that there exists $α>0$ such that $\mathbb{E} |A|^α=1$ and the tail of $B$ is regularly varying with index $-α<0$. We also find the second order asymptotics of the tail of $X$ when $Ψ(x)=Ax+B$.