We know that Snape was tailing Quirrell throughout The Philosopher's Stone, trying to stop him from stealing the Stone. He did this on Dumbledore's orders. But is it possible to say what Snape's (and, by extention, Dumbledore's) motives were here? Was he aware of the bigger picture that Voldemort was trying to steal the Stone through Quirrell? Or was he unaware that Quirrell was possessed, and so thought that Quirrell was acting alone?
I know that he claimed to fellow-Death Eaters that he was unaware that Quirrell was possessed.
“I think you next wanted to know,” he pressed on, a little more loudly, for Bellatrix showed every sign of interrupting, “why I stood between the Dark Lord and the Philosopher's Stone. That is easily answered. He did not know whether he could trust me. He thought, like you, that I had turned from faithful Death Eater to Dumbledore’s stooge. He was in a pitiable condition, very weak, sharing the body of a mediocre wizard. He did not dare reveal himself to a former ally if that ally might turn him over to Dumbledore or the Ministry. I deeply regret that he did not trust me. He would have returned to power three years sooner. As it was, I saw only greedy and unworthy Quirrell attempting to steal the stone and, I admit, I did all I could to thwart him.”
(Half-Blood Prince, Chapter 2, Spinner's End).
Just because Snape claims not to have known Voldemort was possessing Quirrell doesn't mean that he didn't actually know at the time. We know that he pulls the wool over the eyes of other Death Eaters all the time. This could be one such occasion. Claiming that he didn't know Voldemort was involved could just be part of his cover act.
He and Dumbledore would have had plenty of reasons to try and stop Quirrell, even if they didn't realise that he'd teamed up with Voldemort. And Snape suspected Quirrell for the whole year, even before Halloween.
Unfortunately, while everyone else was running around looking for it, Snape, who already suspected me, went straight to the third floor to head me off — and not only did my troll fail to beat you to death, that three-headed dog didn’t even manage to bite Snape’s leg off properly.
(Philosopher's Stone, Chapter 17, The Man With Two Faces).
Harry makes the connection that Voldemort is involved after his encounter with the hooded Quirrell in the Forbidden Forest. Snape didn't benefit from this revelation, however, since Harry thought he was the one who was after the Stone.
Was Snape acting out of ignorance or a wider strategy? Did he think that he was stopping Voldemort or merely stopping Quirrell?