  We show that the loop $O(n)$ model on the hexagonal lattice exhibits
exponential decay of loop sizes whenever $n> 1$ and
$x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)$, for some suitable choice of
$\varepsilon(n)>0$.
  It is expected that, for $n \leq 2$, the model exhibits a phase transition in
terms of~$x$, that separates regimes of polynomial and exponential decay of
loop sizes. In this paradigm, our result implies that the phase transition for
$n \in (1,2]$ occurs at some critical parameter $x_c(n)$ strictly greater than
that $x_c(1) = 1/\sqrt3$. The value of the latter is known since the loop
$O(1)$ model on the hexagonal lattice represents the contours of the
spin-clusters of the Ising model on the triangular lattice.
  The proof is based on developing $n$ as $1+(n-1)$ and exploiting the fact
that, when $x<\tfrac{1}{\sqrt{3}}$, the Ising model exhibits exponential decay
on any (possibly non simply-connected) domain. The latter follows from the
positive association of the FK-Ising representation.
