J.K. Rowling doesn't elaborate much on the Philosopher's Stone at Pottermore [SCREENSHOT 1 || SCREENSHOT 2] The main reason why we don't see more than one Philosopher's Stone in canon during Voldemort's and Harry's journey is that Nicholas Flamel was the only verified Alchemist who successfully created a Stone for 665 years prior to Harry's confrontation with Voldemort/Quirrell (it says others were rumored to have created a Philosopher's Stone, but Flamel seems to be the only named creator in Potterverse):
‘Nicolas Flamel,’ she whispered dramatically, ‘is the only known maker of the Philosopher’s Stone!’ (Hermione)
There have been many reports of the Philosopher’s Stone over the centuries, but the only Stone currently in existence belongs to Mr Nicolas Flamel, the noted alchemist and opera-lover.
Philosopher's Stone - page 163 - UK - chapter 13, Nicholas Flamel
In Tales of Beedle the Bard, Albus Dumbledore wrote in regards to the Resurrection Stone:
Many critics believe that Beedle was inspired by the Philosopher’s Stone, which makes the immortality-inducing Elixir of Life, when creating this stone that can raise the dead.
Tales of Beedle the Bard - page 95 - The Tale of the Three Brothers
The reason Voldemort wasn't interested in procuring another Philosopher's Stone is because Voldemort rejects being dependent upon anything other than himself for immorality -- for anything, really. This is a theme about Voldemort throughout the series. I'm sure the time he spent as practically an infant with Wormtail was intolerable for Voldemort. That's why he chose Horcruxes. They were his; nobody else was involved in their creation or maintenance.
Conversely, if something went wrong with the Elixir of Life -- it was spilled, contaminated, stolen -- then Voldemort would have been just as vulnerable to death as any other mortal. And Voldemort was too arrogant to consider that 1) anyone would guess his secret or 2) that anyone would be able to destroy Horcruxes. He was too arrogant to think he would need a back-up plan.
You can read Dumbledore's explanation of this in chapter 23 of Half-Blood Prince - Horcruxes. It's right after the part where Dumbledore shares with Harry what he knows about Voldemort's Horcruxes. Page 469 in my book, but it's the UK hardcover.
ETA: Pursuant to your comment, I'll add the quote from Dumbledore. As far as I know, there is no further canon on why Voldemort did not attempt to make a Philosopher's Stone himself, or with Wormtail's assistance:
‘While the Elixir of Life does indeed extend life, it must be drunk regularly, for all eternity, if the drinker is to maintain his immortality. Therefore, Voldemort would be entirely dependent on the Elixir, and if it ran out, or was contaminated, or if the Stone was stolen, he would die just like any other man. Voldemort likes to operate alone, remember. I believe that he would have found the thought of being dependent, even on the Elixir, intolerable. Of course he was prepared to drink it if it would take him out of the horrible part-life to which he was condemned after attacking you, but only to regain a body. Thereafter, I am convinced, he intended to continue to rely on his Horcruxes: he would need nothing more, if only he could regain a human form. He was already immortal, you see ... or as close to immortal as any man can be.' (Albus Dumbledore)
Half-Blood Prince - pages 469-470 - UK - chapter 23, Horcruxes
It may also have been that Voldemort didn't know how to make a Philosopher's Stone. Yes, he is a brilliant wizard, but even the most brilliant individuals can't know everything. According to Pottermore [SCREENSHOT 3], the study of Alchemy was not compulsory or even a given elective. Alchemy instruction occurred if a certain number of students wished to study it, during only their sixth and seventh years. I'm willing to wager that Nicholas Flamel had more than two years instruction in Alchemy before creating the Philosopher's Stone.