Introducing supernatural elements mid-work

You ever have one of those ideas that you kick around in your head for ages and it all makes perfect sense to you, but then when you actually sit down and think about it you realize that it wouldn't work for anyone who can't read your mind? ... yeah.

So I've got this cheesy, stupid SF story that goes something like this:

Main Character lives in Dystopian Future Society and is part of a group of young "rebels/revolutionaries" who have a lot of fine ideals but don't in fact do much (or have much practical knowledge of what they should be doing). When they finally decide to make their stand, they are quickly broken up by the police. The MC's buddies all get away, but the MC is captured and carted off to a high-security prison in a really remote location. Sharing a cell with her is another girl who's committed some really minor offense, the kind of thing that would usually only get you a fine or something. MC is able to deceive herself, at least at first, into thinking that she's a big enough threat to merit this kind of reaction, but is really puzzled about why the other girl's there; sure, the government's horrible and arbitrary and mean, but they wouldn't waste time/effort/cell space on this girl unless it was benefiting them somehow.

Long story short, MC eventually learns that most of the people in this prison are there because they've been determined to have latent potential for shiny awesome psychic abilities; they're being experimented on to try to activate the abilities so the government can use them as weapons. As you do. See, I told you it was cheesy.

Anyway, problem is, that reveal comes something like a third of the way into the story. Before that, there's really no mention of psychic abilities potentially existing, because the whole thing's a Top Secret Government Project that ordinary people don't know about. So I imagine a certain number of readers who thought they were getting straight-up non-supernatural dystopian sci-fi action would find this... hard to swallow. On the other hand, if I set it up too much early on, then the audience will figure it out long before the protagonist does and will thus have little patience for her inability to figure it out.

Is there a way to set this up enough that it doesn't come out of left field while still being subtle enough that most readers won't immediately figure out where I'm going with this? Or should I just forget about all this nonsense?

Obvious solution is subtle forshadowing by having the MC accidentally use her powers before, in a way that something strange happens, but there is a logical explanation ready. Repeat two or three times, and you're set.

... oh jeez, how did I not think of that one? Thank you!