We study the susceptible-infected-recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. For this model, each vertex infects all its susceptible neighbors and recovers the day after it was infected. When a single individual is initially infected, the total proportion of individuals who are eventually infected approaches zero as the size of the graph grows towards infinity. Using different scaling, we prove process level scaling limits for the number of individuals infected on day $h$ on the largest connected components of the graph. The scaling limits are contain non-negative jumps corresponding to some vertices of large degree, that is these vertices are super-spreaders. Using weak convergence techniques, we can describe the height profile of the $α$-stable continuum random graph (Goldschmidt et. al. 2018, Conchon-Kerjan - Goldschmidt 2020), extending results known in the Brownian case (Miermont - Sen 2019). We also prove abstract results that can be used on other critical random graph models.