The Moon is generally believed to have formed from debris ejected by a large off-centre collision with the early Earth. The impact orientation and size are constrained by the angular momentum contained in both the Earth's spin and the Moon's orbit, a quantity that has been nearly conserved over the past 4.5 billion years. Simulations of potential moon-forming impacts now achieve resolutions sufficient to study the production of bound debris. However, identifying impacts capable of yielding the Earth-Moon system has proved difficult. Previous works found that forming the Moon with an appropriate impact angular momentum required the impact to occur when the Earth was only about half formed, a more restrictive and problematic model than that originally envisaged. Here we report a class of impacts that yield an iron-poor Moon, as well as the current masses and angular momentum of the Earth-Moon system. This class of impacts involves a smaller-and thus more likely-object than previously considered viable, and suggests that the Moon formed near the very end of Earth's accumulation.