Likely just after the first signs of weirdness started to actively show up. Maybe a week or so of 'monsters' followed by her.
Just to elaborate on the weirdness of the island, I sort of have a pseudo-understanding of why things are all wacky and stuff, since I've gone and sewn it into my backstory and all... I was intending on passing it by you first, but I forgot, so lets do that now, see what you think of this idea, if it's at all similar to your own.
The way I've sort of got it in my head, there's some nexus of supernatural energy on/in/under/around the Bermuda Triangle. A wierding battery that pulses with formless energy that, without thought to direct it, just remains in a contained if chaotic state. Sort of like a primordial soup of paranormal energy.
Once humanity stepped into the area, things changed. Humanity possesses a strong, sentient mind, one constantly thinking, even during rest. There are areas of the mind that remain off-limits to conscious thought. These areas exude an energy, similar to... well, think of it like a smoke-filled box with holes in it. The smoke will slowly seep out of the box, polluting the area around the box, but for whatever comes out, something else will go in, be it air or another box's smoke. This smoke - which I'll call a psionic field, for simplicity's sake - interacts with other fields in random ways, occasionally sending particularly sensitive people signals along an otherwise inaccessible sense. This can be seen in the metaphorical sixth sense or intuition, the mind drawing information from one of these areas of the brain as influenced by the way its psionic field is altered or adjusted by happenings in the world around it.
Now, when a psionic field - literally seeping subconscious will - entered the Bermuda Triangle - a place of pulsing paranatural power - the two reacted with each other. The paranatural essence of the Triangle gave power to those more in tune with their locked off areas of their brain, and the psionic field started to collect and 'evolve' the otherwise inert possible 'life forms' in the paranormal primordial soup.
As such, super powers and monsters.
As far as why everything powers up at night rather than during the day, I dunno, I've not really thought on that so much. Likely it has something to do with the sun's energy oppressing the paranatural field, and the moon empowering it. Likely something tied to pagan worship or some high-powered entities that exist in the two celestial objects.
Anywho, Megami is a monster, just like all the rest. She might be more evolved than some, perhaps more powerful in some respects, but she's still another Para (her term for the 'monsters') like the rest of them.
Assuming whatever nasties we battle are formed in a way similar to Megami's own creation, she wouldn't be aggressive towards them initially, unless they were openly hostile with her. If completely alone with other 'aberrants', she would likely leave them be, perhaps converse. After all, they would effectively be siblings of a sort. In the same way that wild animals of the same species would be when around each other.
That said, should she need to fight and/or destroy them, she would do so without issue.
It's her understanding that nothing is truly destroyed in the Para-Enn (what she's going to refer to the general paranatural-energy-meets-errant-psionic-waves thing that made her), it's simply broken down into base components to be utilized at a later time.
I took Movement: Dimensional with her so that she could step into the Para-Enn whenever she needs to, effectively whatever alternate reality the paranormal oddness exists in.
Oh, and side note: Megami is the name she's actually chosen. Megumi is her 'human' name ^^ So if Paras talk with/at her, and they know her name (which they might), that's what they'd call her.