  Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A
natural variant of Grothendieck's Section Conjecture postulates that every
section of the fundamental exact sequence for $X$ which everywhere locally
comes from a point of $X$ in fact globally comes from a point of $X$. We show
that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it
satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and
give the appropriate generalisation to $S$-integral points on hyperbolic
curves. This gives a new "computational" strategy for proving instances of this
variant of the Section Conjecture, which we carry out for the thrice-punctured
line over $\mathbb{Z}[1/2]$.
