We continue the study of the level-set percolation of the discrete Gaussian free field (GFF) on regular trees in the critical regime, initiated in arXiv:2302.02753. First, we derive a sharp asymptotic estimate for the probability that the connected component of the critical level set containing the root of the tree reaches generation $n$. In particular, we show that the one-arm exponent satisfies $ρ=1$. Next, we establish a Yaglom-type limit theorem for the values of the GFF at generation $n$ within this component. Finally, we show that, after a correct rescaling, this component conditioned on reaching generation $n$ converges, as $n\to\infty$, to Aldous' continuum random tree.