A couple of examples spring to mind:
Unicorn blood. As witnessed by Harry in his first visit to the Forest:
“The blood of a unicorn will keep you alive, even if you are an inch from death, but at a terrible price. You have slain something pure and defenceless to save yourself and you will have but a half life, a cursed life, from the moment the blood touches your lips.”
— Philosopher’s Stone, chapter 15 (The Forbidden Forest)
However, like the Elixir, this seems to be something that only delays death; you need to take it regularly. Whether it was Quirrell or Voldemort who drank it; both of them ended up dead.
Time magic. In the battle in the Ministry of Magic, we see a Death Eater whose head is trapped in a belljar, and his head goes back and forth from being a baby to an adult. If you were to climb inside the jar entirely, you’d regress to being a child, and presumably gain the years back. Of course, this wouldn’t be very useful unless you preserve your adult memories in the process, but it is technically a way to get immortality.
And a few honourable mentions:
Some potions. Harry’s first Potions lesson includes this rather intriguing line:
“I can teach you how to bottle fame, brew glory, even stopper death – if you aren’t as big a bunch of dunderheads as I usually have to teach.”
— Philosopher’s Stone, chapter 8 (The Potions Master)
I don’t recall the specifics of this potion being discussed anywhere in canon. Indeed, there’s no guarantee that it really exists – perhaps Snape was just trying to impress his students – but it’s one to consider.
Edit: I initially interpreted this as meaning “stop somebody from dying”, but other people disagree – see the comments. I think it’s ambiguous, which is why it gets an honourable mention.
Phoenixes. You never specified that it had to be human life. ;)
Anyway, phoenixes seem to have natural immortality (or at least as close to immortality as we see in Potterverse). Since its immortality comes from self-combustion and regeneration, a form of immortality that doesn’t appear in nature1, I assume that phoenix immortality is at least partially magical in nature.
Finally, there are hints that other methods may exist. Dumbledore tells Harry that at sixteen, “Tom Riddle was doing all he could to find out how to make himself immortal”, and at his rebirthing ceremony, he talks about “experiments” into immortality. We only see the one that worked – his horcruxes – but we don’t know whether there were any other plausible candidates.
1 There’s a species of jellyfish, Turritopsis nutricula, which has functional immortality: when it’s dying, it reduces to a small blob until the danger has passed, then regenerates. But this isn’t as impressive as bursting into flame.
