No one notices first-years. Of course there's that big ceremony, the hat shouting at the top of its brims the House it has selected for you, the welcoming cheer at the House table, and so on. But when you run that through an entire class of first-years, the House designation of any individual first-year becomes forgettable. The House system primarily works by self-policing.
Phil Coulson takes advantage of this as soon as possible, so that he can begin establishing a presence in each House. He's not the only one who slips between Houses. There's a "Slytherin" girl, Romanov, whom he knows was sorted into Hufflepuff, because he took out a quill and parchment at the Sorting and surreptitiously recorded every last name and designation. He brings it up to Headmaster Fury. Headmaster Fury tells him not to worry about it. So he doesn't.
Headmaster Fury approves of Coulson's House-swapping – Fury's the one who encouraged it in the first place. Headmaster Fury wants to know every piece of information Phil can grab him, and Phil does, because it takes Phil precisely seventeen days to do enough research and investigation to know that Fury is one of the few Headmasters in Hogwarts history to really have everything under control, and it takes Phil precisely seventeen seconds to know that that is necessary to keep the students from being eaten alive. The last Headmaster had held a major battle of a war at Hogwarts and gotten a number of students killed. Phil is 84% certain that Fury isn't that stupid, which is a high rate of trust for Phil to give anyone.
It isn't long before Phil realizes that Fury isn't just having Phil keep tabs on things. Fury has Phil swap into Ravenclaw the night Stark unveils his newest and most dangerous offering of "prank" weapons for the ongoing Ravenclaw-Slytherin wars. (Every House, Phil has learned, has a war on with Slytherin, including Slytherin itself.) Phil barely even has to think before he finds himself quietly casting curses and hexes the way of the most dangerous tools, because it took one conversation with Stark for Phil to realize that, left unchecked, the boy would turn the entire school to ashes within the course of an evening, or at least give everyone else the tools to do so. Stark has yet to learn that the weapons he makes won't necessarily get used for the things he wants.
After that incident (and getting caught out by Stark, and in true Ravenclaw fashion managing to successfully argue that if Phil could disable them, so could the Slytherins, so obviously Stark needs to build them better, after which Phil realizes that was the worst argument to make because now Stark will build them better), Phil finds himself quietly putting out fires, both literal and figurative, everywhere around the school, with Fury subtly guiding his way.
The Gryffindor Odinson needs to be kept from 1) getting all the other first-years trashed, and 2) starting a real, proper war with Slytherin, weapons and all. The Slytherin Odinson has a new conspiracy every week, and as much foiling as Phil manages, he still comes out at the end of each Slytherin night wondering if it's him that's been foiled instead. So-called Slytherin Romanov is Up to Something, but since Fury has told him to leave it alone, Coulson instead does what he can to prepare for when it all blows to smithereens. The Hufflepuff Barton, meanwhile, just needs to be kept from boredom, because boredom is how love-charmed arrows ended up causing the portrait of the Fat Lady to become desperately enamored with the Bloody Baron.
The worst incident comes on a Tuesday. "Why don't you see your Ravenclaw friends tonight?" leads to a night of Phil thinking he should be keeping an eye on Stark. The mistake is worsened by Stark's insistence on continuously doing brilliantly dangerous things, which in this case involves granting sentience to a chair cushion, and keeping it from smothering Stane is so distracting that Phil misses the chance to prevent real disaster. Because, near the fireplace, quiet little Bruce Banner (lycanthrope, but keeps to himself and never needs a sharp eye) mixes together a Wolfsbane alteration that explodes and turns Banner into a giant green warg wolf.
Stark distracts the monster with enchanted furniture while Phil evacuates the common room. This marks the first time that Phil realizes Stark can be an asset.