For
d
≥
2
d \geq 2
and
n
∈
N
n \in \mathbb {N}
, let
W
n
\mathsf {W}_n
denote the uniform law on self-avoiding walks of length
n
n
beginning at the origin in the nearest-neighbour integer lattice
Z
d
\mathbb {Z}^d
, and write
Γ
\Gamma
for a
W
n
\mathsf {W}_n
-distributed walk. We show that in the closing probability
W
n
(
|
|
Γ
n
|
|
=
1
)
\mathsf {W}_n \big ( \vert \vert \Gamma _n \vert \vert = 1 \big )
that
Γ
\Gamma
’s endpoint neighbours the origin and is at most
n
−
1
/
2
+
o
(
1
)
n^{-1/2 + o(1)}
in any dimension
d
≥
2
d \geq 2
. The method of proof is a reworking of that in [Ann. Probab. 44 (2016), pp. 955–983], which found a closing probability upper bound of
n
−
1
/
4
+
o
(
1
)
n^{-1/4 + o(1)}
. A key element of the proof is made explicit and called the snake method. It is applied to prove the
n
−
1
/
2
+
o
(
1
)
n^{-1/2 + o(1)}
upper bound by means of a technique of Gaussian pattern fluctuation.