When did Tom Riddle get his new face?
Well, I think the confusion comes with mixing movie canon with book canon. In Philosopher's Stone, here's what Harry sees when Quirrell unwraps his turban:
Harry would have screamed, but he couldn’t make a sound. Where there should have been a back to Quirrell’s head, there was a face, the most terrible face Harry had ever seen. It was chalk white with glaring red eyes and slits for nostrils, like a snake.
Philosopher's Stone - page 212 - UK - chapter 17, The Man With Two Faces
So it seems at some point between leaving school, and
sticking his face to the back of Quirrel's headpossessing Quirrel he got a new face some how.
How did he manage that? Is it some dark magic he applies? Is it from possessing snakes in Albania?
In the chamber of secrets when Harry stabs the diary the second time it rips through Tom's face, mainly from his nose. hmmm
Here's a bit from Half-Blood Prince:
Harry let out a hastily stifled gasp. Voldemort had entered the room. His features were not those Harry had seen emerge from the great stone cauldron almost two years before; they were not as snakelike, the eyes were not yet scarlet, the face not yet masklike, and yet he was no longer handsome Tom Riddle. It was as though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of the eyes now had a permanently bloody look, though the pupils were not yet the slits that Harry knew they would become. He was wearing a long black cloak and his face was as pale as the snow glistening on his shoulders.
The Dumbledore behind the desk showed no sign of surprise. Evidently this visit had been made by appointment.
‘Good evening, Tom,’ said Dumbledore easily. ‘Won’t you sit down?’
‘Thank you,’ said Voldemort, and he took the seat to which Dumbledore had gestured – the very seat, by the looks of it, that Harry had just vacated in the present. ‘I heard that you had become Headmaster,’ he said, and his voice was slightly higher and colder than it had been. ‘A worthy choice.’
Half-Blood Prince - page 413 - British Hardcover - chapter 20, Lord Voldemort's Request
I briefly touched on this in What Order Were the Horcruxes Made In. According to the HP LEXICON, Dumbledore was made headmaster of Hogwarts circa 1956. Voldemort appears to be acknowledging Dumbledore's appointment as if it were relatively recent to their visit, so I think it's safe to estimate Dumbledore's memory in the pensieve in Lord Voldemort's Request occurred around circa 1956 - 1957. Voldemort had resurfaced after having gone missing for ten years, immediately following the murder of Hepzibah Smith circa 1946.
In Hokey the House-elf's memory of Tom Riddle meeting with Hepzibah Smith, where he covets the Slytherin locket and the Hufflepuff cup, Harry notices flashes of scarlet in Tom Riddle's eyes here and there, so if you count that as part of his face changing, then it can be demonstrated as early as circa 1946. This scene is also in chapter 20 of HBP, Lord Voldemort's Request.
I postulate that Voldemort's face had begun changing between 1946 and 1956 or 1957, and continued to morph into the terrible visage it became over subsequent years.
The reason for the change in Voldemort's features was his complete immersion in dark, evil magic, including the creation of five Horcruxes by the time he met with Dumbledore in 1956-ish, and the repeated splitting of his soul through murder. Why did he happen to look like a snake? I'm guessing because he was a Slytherin, a Parselmouth, and seemed to admire snakes, as evidenced by his ownership of Nagini. Also, snakes are stealthy, self-sufficient, and potentially lethal (depending on the variety of snake). It may also have been a nod to his ancestry, Voldemort being the last direct descendant of Salazar Slytherin.