Yes, "się" can go to the end.
I'm not sure if this will be the best explanation (and I hope someone gives a better one), but let me try. Usually you'd put "się" in the middle if you're a direct participant in some situation and at the end if you're just describing it from a distance as in - you're not directly involved in it.
For example, let's use the sentence "The child has fallen". If you're witnessing this situation right now, you could call for help saying "Pomocy! To dziecko się przewróciło!" pointing at the child, but if you would report about it, let's say, later in court, you (or the judge) would say "Dziecko przewróciło się, następnie przyjechała karetka" (The child has fallen, then the ambulance arrived).
You would usually hear such sentence structure (with "się" at the end) when watching some documentary on Animal Planet where they would describe the situation as a third-party and say things like "Na widok nadchodzącej gazeli, krokodyl zanurzył się" (At the sight of an approaching gazelle, the crocodile dove under).