We investigate the use of hierarchical phrase-based SMT lattices in end-to-end neural machine translation (NMT). Weight pushing transforms the Hiero scores for complete translation hypotheses, with the full translation grammar score and full n-gram language model score, into posteriors compatible with NMT predictive probabilities. With a slightly modified NMT beam-search decoder we find gains over both Hiero and NMT decoding alone, with practical advantages in extending NMT to very large input and output vocabularies.