This paper presents comparative experimental results for the single-stage and two-stage counter-current acid leaching of battery masses, with the addition of a booster, from different types of LIB waste. Three different types of battery masses were used in this research: Material I, module car; Material II, tablets and laptops; and Material III, mobile phones. These materials were obtained during the mechanical processing of Li-ion battery waste, which were dried at a temperature in the range of 80–180 °C. Leaching studies of these materials were carried out using the single-stage acid leaching method with the addition of hydrogen peroxide, and the innovative two-stage counter-current acid leaching method, also with the addition of hydrogen peroxide. The single-stage leaching of the battery mass (regardless of the composition of the material) in a 15% or 20% sulfuric acid solution with the addition of 30% H2O2 aqueous solution, for 2 h, with a solid-to-liquid-phase ratio of 1:5 or 1:4 at a temperature of 60 °C ensures the leaching of cobalt, nickel, copper and lithium with efficiencies above 95%. On the other hand, the use of an innovative method of two-stage counter-current leaching of the battery mass ensures the leaching of cobalt, nickel, copper and lithium at a level significantly greater than 95%, while obtaining a concentration of cobalt in the leaching solution at a level of nearly 50 g/dm3. It also reduces the leaching time of a single stage to 1 h and, importantly, reduces the amount of waste solutions and the consumption of H2O2 and sulfuric acid. The developed method of the two-stage counter-current leaching of battery masses is therefore characterized by high efficiency and low environmental impact, thanks to which it can be used in commercial processes for the recycling of lithium-ion batteries.