He patrols the 4 hallways in a set and predictable pattern.
When he goes down the hallway that leads back to his cage, that's when he will take the longest to come back.
I ended up figuring that out for myself like 5 minutes after I posted this lol; I moved up to the door on the right before the X intersection and waited until he was out of sight before booking it. The first time I tried I guess I just wasn't far enough inside the door because it didn't close and he crossed the threshold and got me. Made it past that and all the way to the new Phantom Hunt, and I admit I'm now confused here because I'm certain I've explored everything I can and I haven't found a single Phantom yet, just the Ghost Futas. I'll probably find the Phantom eventually, certainly by the time the next update releases at least lol.