Heat ⅓ cup oil in a large pot over medium-low. Cook reserved ribs and stems, stirring often and adding a splash of water if they start to brown, until tender, 5–7 minutes. Add garlic; continue to stir for about 1 minute. Add reserved chard leaves by the handful, letting them wilt before adding more; cook, tossing, until all the leaves are wilted and tender, 10–12 minutes total. Let cool. Squeeze excess liquid from mixture into a measuring glass. (You should have about ½ cup liquid.)