In this article we study a \emph{non-directed} polymer model in dimension $d\ge 2$: we consider a simple symmetric random walk on $\mathbb{Z}^d$ which interacts with a random environment, represented by i.i.d. random variables $(ω_x)_{x\in \mathbb{Z}^d}$. The model consists in modifying the law of the random walk up to time (or length) $N$ by the exponential of $\sum_{x\in \mathcal{R}_N}β(ω_x-h)$ where $\mathcal{R}_N$ is the range of the walk, \textit{i.e.} the set of visited sites up to time $N$, and $β\geq 0,\, h\in \mathbb{R}$ are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking $β:=β_N$ vanishing as the length $N$ goes to infinity, and in the case where the random variables $ω$ have a heavy tail with exponent $α\in (0,d)$. We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes $β_N = \hat βN^{-γ}$ with $γ\geq 0$: we find the correct transversal fluctuation exponent $ξ$ for the polymer (it depends on $α$ and $γ$) and we give the limiting distribution of the rescaled log-partition function. This extends existing works to the non-directed case and to higher dimensions.