Differences between "just might" and "might just?"

"It might just work" more sounds like positive and pragmatic Eg: You are working on a research and you finally say " It might just work" seems like you are hopeful & have some sort of surety

"It just might work" sound more as a natural occurrence & skeptic. Eg: You are stuck with your car left not working you say "it might just work" a little less hopeful plus relying on luck to work it out for you.

I think It might just work could also suggest the meaning "It might work, but barely". With the annotation that the purpose will probably be fulfilled, but only that and nothing more.

We just might have to use both phrases in different sentences; we might just find a clearer—or better—answer.

X: I have already done everything to make my Internet connection faster.
Y: You just might have to turn off your router to fix the problem.

X: What are you doing tonight after your shift?
Y: I might just drop by my parents' house.

X: Are you attending Mia's wedding next month?
Y: I'm not sure yet. I might just go to the reception.

X: It would be more fun if you meet us at the mall, do some window-shopping, and check out the theatre.
Y: Hmmm. I'm not making promises. I just might turn up.

X: I wish I could see ghosts.
Y: Be careful what you wish for. You just might get it.

Sorry, but it's much simpler than all that. To illustrate, I'll address (A) "We might just make it" versus (B) "We just might make it" (for instance, to our destination, given a low tank of gas).

(A) "I think we'll make it . . . but barely."

(B) "Being more optimistic than I normally am, I think there's a chance we'll get there."

A similar bugaboo is (A) "just can't" versus (B) "can't just":

"You just can't do that!" ("Doing that is categorically wrong. Unacceptable. I/we won't stand for it.")

"You can't just [ ]" (sit there; show up; vote; adjust the yaw, but not the roll and pitch.) More is required; what you've done is incomplete, lazy, half-assed.