Solid waste from the decommissioning of coal-fired power plants collected from a power plant in Spain (Puertollano, Ciudad Real) was subjected to acid leaching tests in columns to evaluate the leachability of several valuable and toxic metals (Al, Ca, Fe, K, Mg, Na, Ti, V, Cr, Mn, Ni, and Zn). First, the contaminated waste, delivered by a national company, was chemically characterized. Second, column-leaching tests were conducted using two different acid solutions (nitric and sulfuric acid). The effect of the leaching agent concentration and time of leaching were examined. The results of column leaching tests showed that different concentrations of the acid solutions leached different proportions of Al, Fe, Mg, Mn, Ni, V, and Zn, which were leached by acid solutions from the solid waste sample. In general, use of sulfuric acid at pH 0.5 resulted in better leaching. Next, a comparison between three different configurations (one single stage without recirculation, one single stage with total recirculation of leachate and leaching in two consecutive stages: one with total recirculation of leachate and another one with acid set to a pH value of 0.5 and without recirculation) was performed. At the end of the experiments, all leaching methods resulted in comparable yields for Al (0.36–0.48%), Fe (5.99–6.40%), Mg (4.43–5.11%), Mn (2.71–2.83%), Ni (12.08–12.75%), V (0.08–0.34%), and Zn (23.62–25.28%). However, better results were obtained when two consecutive stages were carried out. Additionally, the effect of forced aeration on leachability was studied. Finally, this investigation showed that hydrometallurgical treatment of contaminated solid by means of acid leaching followed by basic leaching and a water wash between these stages was a potentially feasible method for reducing hazardous levels of the residue.