I don't think we know much about the "Melanesian"^1 dispersal because it happened so long ago - at least 15,000 years, compared to ~2500 years for Polynesian - and most of the Pacific isn't very conducive to preserving archaeological or palaentological remains. That said, remember that much of the peopling of the Pacific by 'Melanesians' happened during Ice Ages when sea levels were low. In those days, Australia and New Guinea were one super continent known as Sahul. It would have been much easier to island-hop in those days from New Guinea, to New Britain, down through the Solomons to Vanuatu and New Caledonia with potentially very simple technology.

^1 "Melanesia" is a racial-geographic classification which is increasingly considered racist and out of date. Many Melanesians are Austronesians as well, since 'Austronesian' refers only to people who speak Austronesian languages, i.e. it's a linguistic classification. These days the distinction between Near and Remote Oceania is becoming increasingly more popular.