Identify what is frustrating about being in an N+1 incarnation (that is, aware that you are living sequential lives). Build from there. It might be:
- Trying to resolve normal and consensus consciousness with your experience of a reincarnated life.
- A wish to send a message to your former self (How might you do this?)
- A wish to send a message to your future incarnation (How might you do this?)
- The sense of impossibility of it all ... that we can never escape this mortal coil
- The sense of wanting desperately to find the answer so that we do not have to repeat another round, but can instead graduate.
- The idea that everyone around you is also reincarnated, and you cannot really explain this to them, because they might think you're nuts.
- The fear that you are nuts.
- Wondering if you were sent back because you didn't 'finish' last time around (overlap with 5 but implying that reincarnation is a result of an accident.)
- Wondering what other paranormal etc things might be true.
- Frustration that what is known about the actual history of who you were last time - - - is wrong. (Imagine Marilyn Monroe seeing all the stuff that had been said about her).
- etc. (or start a list about what is great about reincarnation, the idea of immortality).
So, from those, pick a few that are not interesting to you and throw them out. Figure out a few you might run with - - - as a motivational angle for your character. Maybe some with inherent tension between them. Example, #6 has a lot of interesting potential to me, but a few others do too.
My thoughts on your suggested scenarios:
Being killed by a friend, especially if the friend has also died, leads to the conflict of wanting to find the incarnated friend... Yes?
Being killed by a friend on the battlefield is less compelling because there is no tension in the death - battlefield is naturally dangerous.
Pre/post apocalypse is beside the point. You are focusing on the setting instead of the tension and characters and motivation.
This one is murky to me, not sure how you'd build it out.
Answer: You are going to need to build your story through the choices that your protagonist makes. That may arguably be most interesting to the reader if the story touches on 'big questions' like the reasons for existence: Why are we here? Is there anything else?
My suggestion is to start your story from the other end, decide why reincarnation is interesting/compelling and work the story so you can explore those themes most fully.