The life cycle of dust in the interstellar medium is heavily influenced by outflows from asymptotic giant branch (AGB) and red supergiant (RSG) stars, a large fraction of which is contributed by a few very dusty sources. We compute the dust input to the Small Magellanic Cloud (SMC) by fitting the multi-epoch mid-infrared spectral energy distributions of AGB/RSG candidates with models from the Grid of RSG and AGB ModelS grid, allowing us to estimate the luminosities and dust-production rates (DPRs) of the entire population. By removing contaminants, we guarantee a high-quality data set with reliable DPRs and a complete inventory of the dustiest sources. We find a global AGB/RSG dust-injection rate of (1.3+/-0.1)x10^-6^M_{sun}_/yr, in agreement with estimates derived from mid-infrared colours and excess fluxes. As in the Large Magellanic Cloud, a majority (66 per cent) of the dust arises from the extreme AGB stars, which comprise only ~7 per cent of our sample. A handful of far-infrared sources, whose 24{mu}m fluxes exceed their 8{mu}m fluxes, dominate the dust input. Their inclusion boosts the global DPR by ~1.5x, making it necessary to determine whether they are AGB stars. Model assumptions, rather than missing data, are the major sources of uncertainty; depending on the choice of dust shell expansion speed and dust optical constants, the global DPR can be up to ~10 times higher. Our results suggest a non-stellar origin for the SMC dust, barring as yet undiscovered evolved stars with very high DPRs.