This is very speculative, but I'm suspecting Kipperlily is using the Ethereal Plane to spy on the Bad Kids and if I'm right I might have figured how she got found out by the rogue teacher!
First, let me explain my paranoid thoughts about the spying. Since ep 3, we are aware that Kipperlily is hearing their conversations, even if we don't know for sure the extent of it. She, just like Riz, must have a crazy stealth modifier + reliable talent, however I don't think she's relying only on that.
So, there's a scene in ep 4 when the Bad Kids roll for perception to spot Kipperlily. Kristen casts See Invisibility and Fabian rolls a nat 20. Brennan describes, only to Fabian, that although he doesn't see Kipperlily, he feels the "twinge of some kind of sense". Very creepy. That implies that she's there but cannot be seen. She could be invisible, however I don't think that's the case because an invisible person could still be perceived through hearing or touch. Also, Kristen would have been able to see her because of the spell.
With that in mind, the paranoid goblin that lives inside my brain is convinced that she's using the Ethereal Plane to spy on the Bad Kids. The Border Ethereal is perfect for infiltration and spywork, since you pass through physical objects and watch everything in the material plane without ppl there perceiving you. There are a couple of ways to access the Ethereal Plane, with spells or items, but I have no ideia abt the specific method she might be using.
Her being in the ethereal plane explains why Fabian, with his nat 20, didn't notice any physical indicatives of Kipperlily but still felt a presence, like some sixth sense. Now, Adaine did use her Third Eye so she could see into the Ethereal Plane within 60 feet and still saw nothing. But that doesn't mean that Kipperlily isn't there since there's a very easy way to circumvent Adaine: Non detection. This is a third level spell that hides the target from divination magic - for 8 hours they "can’t be targeted by any divination magic or perceived through magical scrying sensors". The Rat Grinders, long time haters of the Bad Kids, are for sure aware that Adaine is a divination wizard, they would be fools to spy on the party without casting Non Detection first. She's the motherfucking elven oracle!
Okay, now about the rogue teacher. In ep 3, Siobhan theorizes the rogue professor is the ghost teacher. I think she's absolutely right! We know most teachers of Aguefort and even if the ghost one was among those we havent seen, the Bad Kids would probably have heard about them if they were teaching something like ranger class. It's plausible that the reason for the party having no ideia who they are and not even passing by them in the halls is that the ghost is the rogue professor. After all, no one knows who the hell they are, it's the whole point of their teaching method. And for a ghost it would be really easy to go undetected since they can travel through the Ethereal Plane, beside the insane stealth.
Indulge my conspiracy theories for a minute. Rogue professor = Ghost Teacher and Kipperlily can wander in the Ethereal Plane, the plane of ghosts. Even with the advantage of being on the same plane as the teacher, it would still be a nightmare to find them since they are a pro rogue. In fact, Kipperlily didn't achive that: the rogue teacher found her.
But how did she manage to have the professor find her? I'm sure it wasn't an accident, she's too calculating for that. So, I started thinking about what would I do in her place and came up with the stupidest ideia. It's utterly ridiculous. But it could totally work and the strategy seems kinda Kipperlily's style.
Remember the Ghost Steak? The one Fig tried to eat when she invaded the teacher's lounge in season one? It's the ghost teacher's lunch, and Brennan reestablished its existence in ep 3 when Adaine used Ethereal Sight, explaining that the school wards are porous enough to allow ethereal travel and other stuff.
So if I was Kipperlily, my dumbass plan would go as follows: invade the teacher's lounge through the ethereal plane and hide inside the fridge. It wouldn't be a problem for me since I'm intangible and can pass through stuff, plus the other professors wouldn't see me even if they opened the fridge. Inside, I wait as long as needed, until the Rogue/Ghost Teacher gets hungry. They finally open the fridge to grab their snack, only to find me looking at them from inside the fridge like a lunatic. Mission complete: got found by the rogue teacher and aced junior year!
It's so mundane and stupid and that makes me more convinced that's exactly what happened. It's too funny to not be true.