It does however, state that she believes that it was 'almost certainly' Haffaton.
Seeing as Haffaton's Elves fled the hex, signs point to: No.
It shouldn't make a difference. If she moves first, the Turn starting well after dawn tips Haffaton off. If she moves later, dusk not coming with the end of Turn tips Haffaton off. The alert is raised either way.
She's moved into range of Haffaton's forces. The previous day, she started Turn out of range. The determinations/Predictamancy thing indicates that the Battlespace kicked in because they would
encounter Haffaton. That suggests that Side's can share Turns if they just aren't going to fight. Presumably Cities within a day's march of each other share Battlespace regardless, as the choice to attack is always there.
First of all, this was a very helpful explanation. I now understand that sides can share turns if they are nowhere near each other. This wasn't a possibility that occurred to me, but it simplifies things a bit, and explains the difference between the predictamancy of battlespace, and literal "turns." The world itself somehow knows who goes first in a particular area, based on their proximity and other factors.
However! I must insist that the text does not specifically say that the elves fled the hex, nor does it even casually imply that Jillian believes Haffaton is going first. In fact, Jillian is guessing that it will be Haffaton at all, and she clearly isn't familiar with who is sharing the hex she is currently in, since she didn't encounter them last turn. She also doesn't know for sure there will be an encounter at all, there just 'probably' will be.
The "they" would almost certainly mean Haffaton
is a reference to who she believed she would run into, not to who is necessarily going first.
But that secrecy was now distinctly threatened by the fact that someone else had a turn before Faq's today
This is a separate paragraph and a separate idea altogether. Why would she say 'someone else' if she meant Haffaton? There is no reason to believe that Haffaton would mysteriously go first on this turn, since nothing about their nature has changed in any way. Why would someone go first in combat situations, but later in non-combat situations? I'm not challenging this concept, I'm just wondering if this could be a hint to Erfworld's predictamancy. (assuming you are correct)
Now I realize that tying these two paragraphs together, it's pretty clearly stated that Jillian believes that Haffaton is the side going first, but we have no reason to believe that Jillian is intelligent, or able to make such distinctions in her mind. She knows that turn order changes mean trouble for her, we don't know that she's clever enough to decide what exactly causes that trouble.
In the following sentence/paragraph she says:
This was why she hated coming home.
She hates coming home because of the possibility of detection, which means this is always a danger when she returns. But remember that Jillian and her group note the change in turn order this time as an unexpected event. "We're in for it, I guess." If the enemy's presence always causes her to lose her first-turn status, doesn't that mean they wouldn't find the turn order change worth mentioning, as they did this time?
One final comment, your explanation that the enemy would be tipped off by the change in turn order is a very good one, and I'll accept it until I know something else for certain. However, it's just as plausible to say that Haffaton began to control the hex that Jilian was in the previous turn by moving last, didn't see her, and now they are starting another turn with Jillian inactive and exposed a second time to detection. Thus, no one has been tipped off about anything.
Edit: Remember that every time Jillian goes first, the enemy is tipped off to the presence of another force, especially if they are accustomed to going at dawn. This means that Jillian's assumption that not going first would alert others isn't very intelligent. In fact, if the enemy went first like they always did, they wouldn't be tipped off at all. Ie, only Jillian is aware of a problem when she loses her first turn status, and the same is true for the enemy. This is more evidence that Haffaton, (still going first according to your explanation) would know nothing new at all. If this is really how it worked, Jillian should be relived that she didn't go first, since that would mean any enemy nearby would have no idea that there was something to look for.