In this article we study a non-directed polymer model in dimension d ≥ 2: we consider a simple symmetric random walk on Z d which interacts with a random environment, represented by i.i.d. random variables (ωx) x∈Z d. The model consists in modifying the law of the random walk up to time (or length) N by the exponential of x∈R N β(ωx − h) where RN is the range of the walk, i.e. the set of visited sites up to time N , and β ≥ 0, h ∈ R are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking β := βN vanishing as the length N goes to infinity, and in the case where the random variables ω have a heavy tail with exponent α ∈ (0, d). We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes βN = βN −γ with γ ≥ 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.