Frame Challenge: Did Voldemort know that Dumbledore was the original Secret Keeper of the Headquarters of the Order of the Phoenix at 12 Grimmauld Place?
There's a detail worth noting about the reveal of the Fidelius Charm: Dumbledore was able to reveal the location to Harry while not being there to reveal the location to Harry, by way of written note.
Which means that, before Sirius himself reveals that Dumbledore is the Secret Keeper to Harry Potter, Harry Potter didn't even know who the secret keeper was.
It's entirely possible that, given that the others in the Order of the Phoenix, and especially Sirius, didn't trust Snape enough to tell him - I don't have the text on hand to verify that this stayed the case, but it's entirely possible that Snape himself simply didn't know ahead of Dumbledore's death that Dumbledore was the secret keeper. Maybe Dumbledore himself revealed it to Snape, but...
Would Sirius Black have truly done what James Potter did not, and trust Dumbledore over his friends?
We have to remember that Dumbledore did offer to be secret keeper to the Potters back in the day, and James considered it an insult to not trust his friends with his secret. Which probably would have worked, had it not been Peter Pettigrew whom he entrusted the secret too.
Voldemort and Snape both would know that the last time they had to deal with a Fidelius Charm, there was a presumption that Sirius was the Secret Keeper, even though it was actually Peter Pettigrew who was the Secret Keeper, and nobody had clearly attempted to test that theory (Since Peter Pettigrew was still alive, hypothetically, had the charm still been sort of up for others, revealing that Sirius couldn't reveal the location via truth serum or torture would prove he wasn't the Secret Keeper of the Potters).
Would Sirius Black also consider it an insult to trust Dumbledore as Secret Keeper over Remus Lupin, for example? After all, Sirius died earlier, and the Secret Keeper effect wasn't involved then (Presumably as Snape could reveal that he couldn't reveal the Secret of the Headquarters of the Order of the Phoenix then), so the Order of the Phoenix, while they entrusted Sirius to house them, clearly didn't entrust him to stay safe enough to be the Secret Keeper himself, and likely that he didn't trust himself with it. But as Snape found out in Prisoner of Azkaban, Remus still trusted Sirius enough to hear him out about Peter Pettigrew. And Sirius trusted Remus enough to re-use the Shrieking Shack as a hideout way past their original times there. More than he trusted Dumbledore at the time to listen to his story until he had literally no other choice, awaiting the Dementor's Kiss.
Snape had no way of verifying that, in fact, Dumbledore was the original Secret Keeper, short of trying to reveal the secret after he was dead.
The only time he would have absolute proof that the Secret Keeper was dead and the effect was in place was when Yaxley was able to be side-along Apparated there, at which point, even the trio wouldn't stay there, so it was kind of a moot point.