No, relativity deals only with what coordinates different observers assign to events (where an event is a point in four-dimensional spacetime). Observers disagree as to what time or place something happened, but they all agree fundamentally on what happened. If someone in one frame of reference sees a spaceship being hit by a laser, then people in other frames of references may disagree as to when it happened, or whether it happened before or after other events, but they will all agree on whether it hit the spaceship, whether it destroyed the spaceship, etc.
As an analogy, the Gregorian calendar was introduced in 1582. The previous calendar, the Julian calendar, had been "losing" days at a rate of close to one per century. Since it had been in place for over a thousand years, it was off by several days. So the date had to "jump ahead" several days. Does that mean that there was a period of several days in which you could shoot someone, and they could "miss" the bullet because the day was skipped? Of course not. All that changed was what date was assigned to what day. It didn't change what actually physically happened.
If someone on a spaceship is first going away from Earth, and then turns around and comes back to the Earth, then the co-moving reference frame for the first part of their journey will be different from that of the return trip. 562
So if they want to have a co-moving coordinate system in both parts, they're going to have to switch co-ordinate systems. Since the acceleration from going away to going back is when their velocity changes, and thus when they have to change their co-ordinate system if they want to keep it co-moving (and is the point that introduces an asymmetry between the person on the spaceship and a person on Earth, and therefore resolves the paradox of why the two have different elapsed time), this is a natural time in which to consider the "time difference" to happen, but since the amount of time difference depends on the length of the trip, there are also reasons to consider the time difference to accrue throughout the trip.
If at the turn-around point, the people on the spaceship "synch up" with Earth, they're going to have to move their calendar ahead. Since they can't change their velocity instantaneously, if they continuously adjust their calendar then they won't skip any dates, they'll just move through those dates very quickly. But if they adjust their calendar just once, after the acceleration is done, then there will be dates they skip. But there's an important distinction here: there are dates that they skip, but there aren't any days that they skip. A date is simply a label given to a particular day. Some of those labels will not have any days they are assigned to, and therefore be skipped. The people on the spaceship don't skip any actual time, they just skip ahead in their labeling. So, for instance, they might label one day Sep 30, and the next day Nov 1. There is a month of dates that aren't used, but there isn't an actual month of time that the people "jump over", any more than we jump ahead an hour in actual time for daylight saving time. 947