  For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform
law on self-avoiding walks beginning at the origin in the integer lattice
$\mathbb{Z}^d$, and write $\Gamma$ for a $\mathsf{W}_n$-distributed walk. We
show that the closing probability $\mathsf{W}_n \big( \vert \vert \Gamma_n
\vert \vert = 1 \big)$ that $\Gamma$'s endpoint neighbours the origin is at
most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d =
2$. This result is proved using the snake method, a technique for proving
closing probability upper bounds, which originated in [3] and was made explicit
in [8]. Our conclusion is reached by applying the snake method in unison with a
polygon joining technique whose use was initiated by Madras in [13].
