  We compute the integral Chow rings of $\overline{\mathcal M}_{1,n}$ for
$n=3,4$. For $n\leq 6$, these stacks can be obtained by a sequence of weighted
blow-ups and blow-downs from a simple stack, either a weighted projective space
or a Grassmannian. Our strategy consists in inductively computing all the
integral Chow rings of the alternative compactifications introduced by Smyth
and studied by Lekili-Polishchuk.
