What happens to the egg when you boil it is that the protein coagulates. This is similar to how blood coagulate in air but with eggs this process is caused by temperature rather then exposure to air. When you soft boil an egg you heat up the egg so that the white coagulates but not the yoke. This is both because the white is on the outside and therefore heats up faster and also because the yoke coagulates at a higher temperature then the white. But you still heat up the yoke, just not far enough.
When you start with a cooled down soft boiled egg you still need to heat up the yoke to where it will coagulate. It does not help that it was heated part of the way before as there is no changes to the yoke. So you still need to boil the egg for the full duration needed to hard boil the egg. It may actually take longer because the now hard white does not conduct heat into the yoke as well as raw white.
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