We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $α$ and that the Green's function for the random on ${G}$ exhibits a power law decay with exponent $ν$, in the regime $1\leq ν\leq \fracα{2}$. In particular, this includes the cases of ${G}=\mathbb{Z}^{3}$ for which $ν=1$, and ${G}= \mathbb{Z}^{4}$ for which $ν=\fracα{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\fracν{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $ν>1$ as well as corrections of order $\log \log R$ when $ν=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $ν>1$ and essentially optimal when $ν=1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117.