Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift $μ>0$ and branching according to a law in the domain of attraction of the $(1+β)$-stable distribution. The mean of the branching law is strictly larger than $1$ implying that the system is supercritical and the total number of particles grows exponentially at some rate $λ>0$. It is known that the system obeys a law of large numbers. In the paper we study its rate of convergence. We discover an interesting interplay between the branching rate $λ$ and the drift parameter $μ$. There are three regimes of the second order behavior: $\cdot$ small branching, $λ<(1+1/β) μ$, then the speed of convergence is the same as in the stable central limit theorem but the limit is affected by the dependence between particles. $\cdot$ critical branching, $λ=(1+1/β) μ$, then the dependence becomes strong enough to make the rate of convergence slightly smaller, yet the qualitative behaviour still resembles the stable central limit theorem $\cdot$ large branching, $λ> (1+1/β) μ$, then the dependence manifests much more profoundly, the rate of convergence is substantially smaller and strangely the limit holds a.s.