We extend the Lévy Langevin Monte Carlo method studied by Oechsler (2024): Choosing a heavy-tailed target distribution we prove convergence of a solution of a stochastic differential equation to this target. Hereby, the stochastic differential equation is driven by a compound Poisson process - unlike in the case of a classical Langevin diffusion. The method allows one to sample from non-smooth targets and distributions with separated modes with exponential convergence to the invariant distribution, which in general cannot be guaranteed by the classical Langevin diffusion in presence of heavy tails. The method is promising due to the possibility of a simple implementation because of the compound Poisson noise term.