Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity’s strength (ζ). This observation helped us to uncover a hidden connection between active SOC systems andα-stable Levy systems. We demonstrate that one can partially sweepα-stable Levy distributions by changingζ. The system undergoes a crossover towards Bak–Tang–Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover pointζ<ζ*≈0.1.