After doing a bit of looking into the topic, I think I may have found some sources to answer this question. I'll wait until others comment, to hear what they think, before I select this as the correct answer.
As discussed in this Chabad article and this question, there are five levels of the soul: nefesh, ruach, neshamah, chaya, and yechidah. As mentioned in both of those sources, the neshamah refers to the thinking part of a person, the ruach is the emotional part of a person, and the nefesh is the life force and desires of a person.
When we drift off into the subconscious, we still are able to think on some level. It's for this reason that we're able to interpret the 1/60 of prophecy contained in dreams. This is because not all of the neshamah has left the body - 59/60 of it remains. But in the unconscious state, the brain focuses its energies on the basic needs in order to keep the body alive. There is no thinking done whatsoever. It's like the entire neshamah has left and no thinking is occurring, in order to leave the nefesh to keep the body alive.
Another proof can be brought from Bava Basra 10a:
Yosef the son of R' Yehoshua had been ill and fell into a trance. His father asked him: "What did you see?" He replied, "I saw a world upside down, the upper below and the lower above." He said to him, "You saw a well regulated world."
From the fact that he saw a higher level of Olam HaEmes than one sees in dreams, which reflect but 1/60 of what the neshamah normally sees, it's quite possible that more of his neshamah went up to Shamayim, and thus he was able to bring back a clearer picture.
TL;DR: If I am correct, the entire neshamah leaves the body, and one is all but dead in the unconscious state.
