I suspect Quirrell's closing statements to McGonagall at the end of this chapter are not quite what they seem. I'm thinking of two in particular: the first, that he wants Harry kept away from the Restricted Section, and the second, that he wants McGonagall and Dumbledore to try to restore Harry's mood by any means necessary.

The trick to the first one is that he hasn't mentioned sealing off a certain other means of discovering arcane secrets at Hogwarts. Admittedly, Quirrell's suggested that it's probably blocked off anyways. But it might not be; even if the basilisk itself is gone, there might still be useful books. So it looks as though Quirrell is trying to push Harry into finding the Chamber of Secrets. There could be any number of reasons why - though the fact that it's a secret, hidden place at least partially exempt from the Hogwarts wards seems like a good place to start.

The trick to the second one is that McGonagall's way of cheering Harry up is actually going to be quite predictable: she and Dumbledore are likely going to try bringing Harry's parents to Hogwarts. Naturally, this opens up a whole world of possibilities for Quirrell; he could use them as hostages, kill them, Imperius them, or do any number of other nasty things if necessary. Or, if he's interested in understanding Harry better, he could use Legilimency to learn all about his background.