At CRYPTO 2012, Knellwolf and Khovratovich presented a differential formulation of advanced meet-in-the-middle techniques for preimage attacks on hash functions. They demonstrated the usefulness of their approach by significantly improving the previously best known attacks on SHA-1 from CRYPTO 2009, increasing the number of attacked rounds from a 48-round one-block pseudo-preimage without padding and a 48-round two-block preimage without padding to a 57-round one-block preimage without padding and a 57-round two-block preimage with padding, out of 80 rounds for the full function. In this work, we exploit further the differential view of meet-in-the-middle techniques and generalize it to higher-order differentials. Despite being an important technique dating from the mid-90's, this is the first time higher-order differentials have been applied to meet-in-the-middle preimages. We show that doing so may lead to significant improvements to preimage attacks on hash functions with a simple linear message expansion. We extend the number of attacked rounds on SHA-1 to give a 62-round one-block preimage without padding, a 56-round one-block preimage with padding, and a 62-round two-block preimage with padding. We also apply our framework to the more recent SHA-3 finalist BLAKE and its newer variant BLAKE2, and give an attack for a 2.75-round preimage with padding, and a 7.5-round pseudo-preimage on the compression function.