There is a concept in time-space called the worldline. The worldline of an object is a four-dimensional plot through space and time. Part of the concept of a worldline is to remove the concept of time as moving, and show the world as though you could see everything from all time “at the same time” (not a very elegant description but it’s difficult to explain without pictures).
For a simpler example, imagine a “bug” (a point, really) who lives on a one-dimensional line, moving back and forth. To plot a simple two-dimensional worldline, imagine the y axis is the bug’s position on the line, and the x axis is time. The worldline of the bug would zig-zag up and down but the plot would continuously move to the right. That is, it would be a continuous function; a given value of x would produce exactly one value of y.
Now imagine the bug could travel in time. The worldline is no longer a continuous function but a relation. It can hop backwards or forwards. That implies that the bug can be two places at the same time (but it’s the bug at two different ages) or there can be a time where is does not exist at all.
The bug could jump back in time, hang out for a year, then return to the point where he departed, although now he’d have aged a year, biologically. So there is no gap where it doesn’t exist, but there is an interval where he existed in two places at the same time. His world line has a discontinuity.
I do not know what current physics theories have to say about the possibility of time travel, problems with causality paradoxes (let’s say you go back in time and kill your younger self, you could not have survived to age into the future and go back in time and kill yourself, so you didn’t, but then you *did * survive, and so…), or the possibility of the same matter existing in two places at the same time. But in a science fiction story the author can set his own ground rules (I’m not familiar with the story you’re reading). These questions you are asking have been asked by science fiction readers (and scientists) for decades.
Another problem that is not brought up as often is if you can travel in time, say to back six months, that causes certain paradoxes about *where * you pop out. The Earth has gone halfway around the sun, rotated about 185 times, the sun has been zooming through the galaxy, the galaxy is doing I-don’t-know-what, and all of space-time is expanding. So it’s pretty sure you wouldn’t end up standing in the same spot you were when you pushed the button.