GREEK AND ENGLISH:
Evelyn-White, Hugh G., trans. Hesiod, The Homeric Hymns, and Homerica. LCL. London: Heinemann, 1914.
Murray, A. T., trans. Homer: The Odyssey. 2 vols. LCL. London: Heinemann, 1919.
———, trans. Homer: The Iliad. 2 vols. LCL. London: Heinemann, 1924–1925.
Books 13–24, not available on archive.org.
Note: not all LCL volumes are openly available online. As always, see the Digital Loeb Classical Library for up-to-date texts and indexing. Regarding the above, see particularly the new 2 volume LCL edition of Hesiod (Most 2007), the new edition of the Homeric Hymns (West 2003), as well as the revised versions of both the Odyssey and the Iliad.
Greek Text (Iliad Only):
Allen, Thomas W., and David B. Monro, eds. Homeri Opera. 5 vols. OCT. Oxford: Clarendon, 1920.