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There are several ferry companies accessible from Miyazaki. Traveling on an overnight ferry is nice because you can move about the boat, eat dinner, relax, and drink. There is also the option to bring your car along for the trip, though sometimes just renting a car at your destination is more economical.
The Miyazaki Car Ferry travels from Miyazaki Port to Kobe. You can book your ticket online in Japanese, at the Miyazaki City ferry terminal, or over the telephone in Japanese. This website has an English version, but ferry tickets cannot be booked via the internet in English.
Sunflower Ferry has three routes: Osaka to Oita, Osaka to Shibushi (Kagoshima), and Kobe to Beppu. While none of these are in Miyazaki, depending on where in the prefecture you live they might provide more travel options for you.
If you’re traveling to Shikoku, the Uwajima Unyu Ferry operates a route from Usuki in Oita to Yawatahama in Ehime. The website is only in Japanese, but you can find an English timetable and fare chart here.