So when it arrives, it's clock indicates that perhaps a day has passed
No, nearly a year has indeed passed. From your point of view, EVERYONE ELSE seems to have slowed down.
I don't think that's how it works, at least according to Einstein and Hawking who talk about a starship traveller returning and meeting his great grandchildren. Travelling at high speed is like getting very close to a black hole. Hawking discusses that as a method of prolonging your life so that you can watch the universe evolve over a human lifetime, and that travelling at high warp is a more practical way of doing the same thing.
What you're talking about, possibly, is that the 0.999c traveller would see light
from the people he left behind as slowing down (because the light from them is barely catching up to him). Once he arrives, the light from them that's arriving was that sent barely after he left, so they sseem like they're stuck a year in the past. But they're not, the traveller is just a light-year away, observing their activities from a year behind. His year is actually missing
. It didn't happen (as if he'd been very close to the event horizon of a black hole). The people he's observing will eventually catch up to him traveling at 0.5C, and he'll realize that the apparent one-year lag (where he thought he was in-sync with their timeline) was an illusion created by being 1 light-year away.
Or, more simply, if he travels one light-year at 0.9999C and looks back, the light he's seeing is from shortly after he'd left. To him only a day has transpired, and only about a day on the home planet seems to have transpired, as far as he can tell from light that's a year out of date. If he turns around and goes back at 0.99999C, two days have transpired to him, but two years
have transpired on the home planet he returns to.
That's why Einstein and Hawking talk about time-dilation, or getting very close to a black hole, as a form of time travel.