The path taken doesn't matter. Travel from the Western Hemisphere to the Eastern Hemisphere always moves ahead in time (against the sun). Travel from the Eastern Hemisphere to the Western Hemisphere always moves back in time (with the sun). At the International Date Line, the Western Hemisphere is east of the Eastern Hemisphere and the Eastern Hemisphere is west of the Western Hemisphere.

Ideally everybody would just use Coordinated Universal Time instead so that the time would be the same for everybody no matter where on Earth they are.

Another issue is occasionally leap seconds are added to a minute making it sixty-one seconds long. But when these leap seconds occur is determined on a case by case basis. That means that every clock either has to be kept updated or they will become a second out of sync for every leap second.