Sorry, this is long and winding and ultimately goes nowhere, because it is a deconstruction and not an argument. But I think the perspective is useful.
First, the detail of deep sleep is irrelevant:
If while sleeping I adjust my covers in the middle of the night because I am cold, I am applying logic, I am reacting to a sensation -- I am conscious. I have no ongoing memory stream to go along with that consciousness, but I am not simply making autonomic bodily motions like breathing or reflexive responses like moving away from a touch. My behavior is purposeful and involves both awareness and deduction, if on a rather minimal level. So the argument that consciousness ceases in sleep is unconvincing. Consciousness may reduce to an arbitrarily low level, but there is no proof it ever ceases. Arguments from the 'fact' that we are unconscious during sleep are therefore not really proven.
Second, all the details of timing are irrelevant:
If consciousness inheres in a memory stream then when two copies that proceed from the same memory stream exist, they either share consciousness, or they do not. So if I made a copy of myself, without destroying the original, would that copy share my consciousness, or not? Our instinct is no. The two copies would have separate wills, and could make differing decisions as soon as both existed. If the will is not unified, it is hard to see that will as belonging to an individual. So there are multiple individuals, and multiple consciousnesses.
So if we make a copy, and we consider the consciousness of the copy, how can it matter whether the original is destroyed? For it to matter would be a strange sort of change at a distance without any contact or connection. Consider this in a sort of 'Schroedinger's cat' scenario, where the original is isolated from us at the moment the copy comes into being. If the original were destroyed at just the right time, we would have to say this copy's consciousness was a continuation of the original's and if it were destroyed a single second later, we would have to decide otherwise. Since it is possible for us to not even know, I don't see how we can consider this a form of logic. It is really only an analogy. There is simply an arbitrary choice of labels, and labeling is not an aspect of consciousness. What we see continuing is identity from a third-party point of view, and not consciousness.
In that case, that continuity is the same whether there is only one of me at a time or whether you multiply me. And the prior logic suggests that it is not the same consciousness if you do multiply me. So this is not the same consciousness.
Finally, the concept of continuity is irrelevant:
If I awaken, as the copy, my behavior is a continuation of the memory stream of the original. If I awaken as the original, my behavior is a continuation of the memory stream of the original. If I as the original am killed and reconstituted centuries later, and two dozen copies are made, each of them has behavior that is a continuation of the memory stream of the original. But these will be a second consciousness, or maybe a single consciousness or a dozen consciousnesses. So why do we care about continuity of consciousness?
We don't.
Daniel Dennett's argument is that consciousness is an orienting signal that sits ready to adapt to updates in the memory stream. The timeline of experience is awareness of the changes in our composite mental state as we try to choose the next memory to consider conscious. But we can be conscious of a memory that is seconds old or one that is decades old before we return to the state of seeking a next conscious focus. So consciousness is in fact always of the past, and is indistinguishable from memory itself.
In that case, consciousness of the current moment is impossible. The feeling of it is an epiphenomenon, not a reality, and that is why it is basically impossible to argue about it logically. You can continually choose as your current memory the sensation of being focused on the temporal stream in the last focal period, but by the time you can choose it, even that is a memory of an event in the past.
If 'now' is not a thing, consciousness itself is not continuous from instant to instant, and it does not have or need a timeline. Nor can providing it a timeline unify or distinguish between 'instances of consciousness'. This whole line of reasoning within the philosophy of mind needs a different orientation and a different vocabulary.
