OOTA is great, but it does need (mental) prep. That said you can prep a chapter or two ahead and be fine.
Although I'm sure you can prep just a little bit ahead of where the PCs are currently up to in the adventure and run it just fine, one of the downsides to this is missing out on foreshadowing opportunities.
One of the biggest learnings I had from running my Shackled City AP about 8 years ago now is that foreshadowing events that will happen later in the campaign make for some of the most awesome table moments.
For example, in the SCAP, there is a Beholder that, in the adventure as written, pops up in Chapter 1 then you don't see him again until right near the end. In my campaign I had the Beholder show up multiple times during the campaign, mostly when the PCs were in no position to try and fight it. What that meant though was when the PCs finally did fight and defeat the Beholder it meant so much more.
Similarly, there is a huge red dragon that shows up near the end of the campaign. I foreshadowed it during the whole campaign, from rumours about it having a lair nearby, eyewitness reports of dragon sightings, tales of the legendary dragon's ramapages from the past, etc. Sometimes I'd even just mention that the PCs thought they saw something big flying in the distance, but it was too far away to be sure. When the players finally did encounter the dragon I simply plopped the dragon mini on the table and watched their eyes go wide as they realised that they were about to go up against this creature of legend that they'd heard so much about.
Moments like those, which are still recounted in my group years later, only happended because of the foreshadowing that went on beforehand. Without it, it would have been just another beholder or dragon encounter.
I've also found that the longer the period of time (both in-game and out-of-game) between first foreshadowing the event or incident with the creature and the final encounter, the bigger the impact.