So typically big new FF trailers are generally shown at the Sony conference, but that's not set in stone. You always get publisher/ developer conferences prior to the official start of E3 (and E3 officially opens on Tuesday), So Sony, Microsoft, sometimes Nintendo, EA and Ubisoft always have conferences the day before (this year it's Sunday/ Monday). Last year and this year Bethesda also had a conference, and other devs/ publishers have had their own in the past (Square Enix had one last year, Konami (lol) have had one a couple of times).
The big conferences like that are just announcements and or "this is what we're doing", and they're the thing to watch for.
For the past few years, SE have had "Square Enix presents" which is a livestream that basically runs the length of the whole E3 (and by that I mean when E3 officially starts, which is on Tuesday - Thursday). It's safe to say that if there are any major announcements, it'll be during Sony's conference like it usually is. More than likely (since we generally know all of SE's big upcoming games) they'll use the Square Enix presents channel to talk about and show off new stuff about their lineup of games (with a focus on World of Final Fantasy, Final Fantasy XV, Final Fantasy XIV and Deus Ex: Mankind Divided, since those are all imminent).
This is not to say that they won't surprise us with some new stuff at the Sony press conference tomorrow. For example if they ARE going to show any new FFVII stuff, it'll definitely be at the Sony conference because there's no space for it on their own livestream schedule.
So we have the Big Deal(tm) conferences (Bethesda and EA have been - Sunday - then Ubisoft, Microsoft, PC Gaming Show and Sony on Monday). Then I think there's a Nintendo thing on Tuesday just before E3 properly opens, and it's during that time that SE starts their stream running for the length of the rest of E3 (until Thursday).