I think we're seeing the set up of the end game of the upload of Quirrellmort to Harry.

David Monroe, last descendant of a Noble House, family killed by Voldemort, the great champion to save Magical Britain against Voldemort. But instead of having the mass of Magical Britain rally around him, they generally make his difficult. When that plan of being a savior fails.

He then disappears. Poof.

Where is our champion now?

And poof! Harry Potter, last descendant of a Noble House, family killed by Voldemort, miraculously ends Voldemort rampage.

Years later, Monroe reappears as Quirrell, just in time to be the Defense Professor and mentor for this miraculous Harry Potter, who in fact is miraculous, for his age, or really any age. With a secret Dark Side that gives him great power. With so many similarities to Quirrell in personality and intelligence. With such a natural bond.

Perssonalitiess change, mix with victim'ss. Death iss not truly gainssaid. Real sself is losst, as you ssay. Not to my pressent tasste.

But if the victim is already very similar, perhaps more to his taste?

David Monroe, 2.0. Prepared as a baby to be a host. Prepared with a better history to be Savior of Magical Britain, prepared to be so similar to Quirrell.

We've got Harry bringing up Horcruxes, Quirrell going beyond his usual psycho smirk to burst out laughing, Quirrell explaining that a Horcrux is a ritual that transfers intelligence to an object, but woe is me,

Merlin'ss Interdict preventss powerful sspells from passing through ssuch a device, ssince it iss not truly alive.

If only there were objects that weren't not alive that the intelligence can pass into, then they'd get all the powerful spells too.

Oh, and by the way, I was going to teach you all my powerful spells, but you never asked, and now there is no time, no time.

If only there were a quick way to pass all my knowledge to you, but alas,...

Quirrell has Harry begging for this lost knowledge. Perhaps he'd also be begging to be part of a ritual to pass Quirrell's intelligence into him?

Other things.

Why forbid Harry from bringing him unicorns? For that matter, why didn't Quirrell feast on unicorns again and again to keep his strength up? Why doesn't he have a herd of them up in a ranch in the mountains somewhere?

Harry turned, stared at the surrounding trees. "Have a herd of unicorns at St. Mungos. Floo the patients there, or use portkeys."

"Yes, that would work."

The Sense of Doom is diminishing. Perhaps this is what would allow their magic to interact, and thereby complete the ritual? Perhaps that's why Quirrell isn't more effectively fighting his failing health. Or is that only to motivate Harry to look into methods or fighting death with some desperation?

EDIT
On the "uploading to a new body" idea, see Quirrell rubbing their noses in their ignorance:

The smile that the Auror gave had absolutely no mirth in it. “So where’s the real Quirinus Quirrell, eh? Under an Imperius in the bottom of a trunk somewheree, while you take a hair now and then for your illegal Polyjuice?”
"You are making highly questionable assumptions," the Defense Professor said with an edged voice. "What makes you think I did not steal his body outright using incredibly Dark magic?"