In Homer the Elysian plain is not a reward for virtue. Menelaus is told he will go there because he is married to Helen and is therefore son-in-law to Zeus, and the reason given is kinship. It is an exemption granted by connection, which is how a good deal of archaic Greek privilege worked.
The moralising came later and unevenly. Hesiod places the heroes of the fourth age on the Isles of the Blessed. Pindar, writing for aristocratic patrons, gives a version in which souls who keep their oaths through three lives in each world reach the island of the blessed, which introduces both a moral criterion and reincarnation. Virgil in the sixth book of the Aeneid gives Elysium a population of the pious, the patriotic and the artistically gifted, which is a Roman list.
By the time the idea reaches the schoolroom it has become one third of a tidy scheme with Elysium for the good, the Asphodel Meadows for the ordinary and Tartarus for the wicked. No ancient author sets it out that way. The scheme is a later convenience assembled from authors who were writing centuries apart with different purposes, and it is worth knowing that the neatness is a modern contribution.
The location drifted as well. Homer places it at the ends of the earth by the stream of Ocean, later writers move it to islands in the far west, and Virgil puts it underground alongside Tartarus. A paradise that is over the horizon and a paradise that is below your feet are answering different questions, and the tradition used both without reconciling them.