He does hide it. In the part in the graveyard where he is talking to Snape while they are poisoning Riddle Sr’s grave, he keeps spinning around, despite the fact that the eye lets him see 360 degrees regardless of where it is pointing.
If he’s escorting Potter around, everyone KNOWS he’d be on high alert, so they expect to see the thing whizzing around. Then they expect that when it isn’t whizzing around, it means he can’t see behind him.
And when they try to exploit that, that’s how he catches them.
Well, this chapter is just full of delicious puns, apart from the entire plot advancement thing.
Auxiliary Protective Special Committee.
APSC
Absurdly Powerful Student Council.
Daphne worried that Draco would be skinned and turned into Leather Pants.
Draco returns “at the turn of the tide” wearing white (silver) robes. He’s Draco the White.. err.. Silver.
Considering the short length of the chapter, and combined with the call forward to Book V’s Ministerial Education Decrees, that’s a good number of references.
The real question is where does it go from here.