One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+θvv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with centered standard normal distributions, and $θ>0, v \in \mathbb{S}^{n-1}.$ It is well known $λ_1(P),$ the largest eigenvalue of $P,$ has a phase transition at $θ_0=1:$ when $θ\leq 1,$ $λ_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $θ> 1,$ $λ_1(P) \xrightarrow[]{a.s.} θ+θ^{-1}.$ Under more general conditions, the limiting behavior of $λ_1(P),$ appropriately normalized, has also been established: it is normal if $||v||_{\infty}=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $α\in (0,4),$ the fluctuations are shown to be universal and dependent on $θ$ but not on $v,$ whereas a subfamily of the edge case $α=4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at $θ_0=1, θ_0 \in [1,\frac{128}{89}],$ respectively. These results build on our previous which analyzes the asymptotic behavior of $λ_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.