We add relaxation mechanisms that mimic the effect of temperature and non-equilibrium driving to the recently proposed spiral model which jams at a critical density ρc < 1. This enables us to explore unjamming by temperature or driving at ρc < ρ < 1. We numerically calculate the relaxation time of the persistence function and its spatial heterogeneity. We disentangle the three different relaxation mechanisms responsible for unjamming when varying density, temperature, and driving strength, respectively. We show that the spatial scale of dynamic heterogeneity depends on density much more strongly than on temperature and driving.