“Priam’s Treasure”, or what?
There is evidence that even King Tut’s tomb was looted at least once in ancient times before it was discovered by Carter. Carter concluded that the tomb had been looted twice but only small items of jewelry and oils had been taken.
Khaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaan!
(Seriously - they never found the tomb of the man who conquered and looted half of Eurasia. Gotta be something worthwhile there).
Looking specifically for “discovered” (see OP) unlooted tombs friend. 
–thereby making it not only the greatest tomb hoard in history, but also providing evidence for the Least Observant Tomb Looters in History.
“Inerkha! Bring that torch over here! Yes…YES! OILS! Jackpot!”
“Oh, man… we’re almost out of torches already. Did we check that other corridor over there yet, or–”
“Don’t worry about it, dude; just grab as many oils as you can. What are the odds there’s anything more valuable than this lying around down here?”
Don’t smoke mummy wrappings, kids.
Alexander?
Jimmy Hoffa?
Discovered tombs, not undiscovered tombs.
How can something be the “next discovered”? Either they already found it (in which case it can’t be “next”) or they didn’t.
I think the OP means - which tomb (other than Tut’s) is considered the next-best discovery of an unlooted tomb?
Not a yet-to-be-discovered tomb, but one which has been discovered and is considered the second-best unlooted tomb discovery.
BTW, “Priam’s Treasure” from the OP, doesn’t really qualify. That find suffered badly from Schliemann’s methods; a modern archaeologist would have wanted to strangle him.
I offer the tomb of the Lord of Sipan, in Peru as a possible candidate.
I would nominated the Lord of Sipan, the undisturbed tomb of a Moche warrior priest in Peru.
He is often referred to as the King Tut of the New World.
It might be my Anglophile bias showing, but I’m rather partial to the
Sutton Hoo burial. It is, like the wikipedia article says, the jump-off point for English History (vs. Romano-Brythonic History)
…so why was his tomb so lavishly endowed with golden artifacts? Surely a major pharoah (like Ramses) would have much more loot than Tut. As for the tombs robbed in antiquity-what happened to the loot?
And correspondingly the robbers got to him quicker, due to the greater motivation 
The Pharaoh is the Pharaoh, you give him the best of everything. But that IS part of the reason Tut was undisturbed; he was such a minor figure he was likely overlooked (and IIRC his tomb entrance was obscured by a landslide or something so it was a bit of a schlep to make the effort)
Hey, gold is gold, jewels are jewels. Melt the metal and literally make your own money; sell the spices, gemstones, and art pieces.
I would probably Emperor Chi’in, the first emperor of China. We know exactly where his burial mound is (it was discovered in the mid-1970’s), and we have lots of stories regarding what was buried with him, but so far nobody’s gone inside. The famous army of terracotta warriors are about a mile or so away from the actual tomb.
BTW, to this very day, archaeologists working in Mesoamerica or Perú have to worry that if they make a find and can’t secure it immediately, it can get cleaned out literally overnight; or that once a find is announced, every knoll on the ground in 5 miles around will get destroyed by folks just digging for treasure. Why? Because, gold is gold, jewels are jewels. . .
I would have mentioned Sipan myself, but I have been beaten to it.
In the next few years, I think that there is the chance that some ‘big’ burials will be found in Guatemala at the recently discovered (by legit archaeologists, anyway) Site Q.
There are two ways to look at this. First, was Tut a minor pharaoh? In hindsight perhaps, but to his contemporaries he was their pharaoh. Ever royal family likes to think of themselves as important, surely? Second, because we have no other unlooted pharaoh’s tomb to compare it with, we don’t know that Tut’s was lavish compared to others. For all we know, Tut’s stash may have been minor compared to the tombs of other pharaohs.
No way, man. Tut had an historic stash. What do you think was in those oils, man? Dude, that’s why the grave robbers left all that gold and jewels other non comestibles, they were out of their minds.
Tut’s stash, minor?
It is to laugh.