To speak broadly, the answer is: pretty much any site that wasn't entirely, newly discovered in the 20th century.^(1) If you're looking for sites that were rediscovered and properly excavated around the same time, that almost by definition has to be a site explored in the 20th century, as proper scientific methods of excavation, and the field of archaeology as a whole, didn't really come into existence until that point. There was archaeological exploration in the 18th century and before, but it was often little more than treasure hunting. 19th century excavations were better but generally had little to no documentation, often only kept items deemed worthy of attention (attractive things considered 'art,' as opposed to the vast amounts of things we collect and keep now as artifacts), and may have been done quickly and according to the whims of the individual in charge. There are exceptions to all of these statements, of course, but I am thinking of the excavations of Heinrich Schliemann (late 19th century, at Hissarlik/Troy, and Greek sites of Orchomenos and Mycenae), Sir Arthur Evans (right on the nose in 1900, at Knossos, which he completed in 3 years), and Giuseppe Fiorelli as director of excavations at Pompeii in the late 1800s. Of these three examples, there is a spectrum of what was even then considered responsible (looking at you, Heinrich, and the Treasure of Priam...), versus what was considered meticulous at the time (Fiorelli) but by modern standards their work doesn't scratch the surface of what we do now.

Full disclosure: I am a classical archaeologist, so my answer will be heavily tilted toward sites from the Mediterranean; I do hope someone trained in the archaeology of other places will weigh in!

Pompeii is a good example of exactly the phenomenon you mention: it was first identified in the late 16th century through the digging of a water canal. Its excavation didn't begin until the 1780s, under the bourbon kings who held Naples at the time, and what we would term scientific excavation didn't begin there until Fiorelli's time. So, this is a perfect example: a site that was fully underground in the modern era, found but not explored until significantly later.

I'm struggling to find other sites that fit this exact pattern, however, because there's an important point to add to your question, and that is who do we consider to have discovered these sites?

I'm thinking specifically of places where the ruins were never fully underground - where one could just wander through the right part of land, turn a corner or come over the crest of a hill, and go "oh my god that is a Greek temple!" or similar. That kind of discovery almost always comes via a foreign visitor, who brings that information back home and then is lauded as a sites's discoverer, though the locals always knew that site was there, and it was never truly lost. As I say this, I am thinking of Paestum (Poseidonia), a city in southern Italy founded ca. 600 BC by the Greeks and later a Roman city, abandoned in the 9th c. AD due to a rising water table and increase of malarial conditions and other issues. Have a look at the three intact Greek stone temples that still stand today; these were never buried. The British engraver Thomas Major visited Paestum in 1767 and published a book the next year of what he saw there: these incredible images show us that this site was always visible thanks to these temples, the fortification walls and the gates. Though this ancient city was abandoned, the residents clearly simply moved to a nearby area - settlements such as Capaccio Vecchio were founded on higher ground, and through the middle ages there is evidence of locals in the area returning to Paestum to remove stone for building material. Pedley also notes that the site was even on sea charts from the 18th century, telling us the site was visible from the sea and could be a landmark for those navigating the waters nearby (Pedley 1990, 17).

However, if you read up on Paestum you'll be told it was rediscovered in the 18th century, when Count Gazzola - a dignitary from the nearby Bourbon court in Naples - visited and then announced he had "discovered" the site upon his return. (To be clear - Thomas Major visited after Gazzola, I just wanted to use his visuals to highlight how obvious this site really was.)

So this site was never truly lost - the locals always knew where it was. But it wasn't until an aristocratic visitor with the right connections showed up and claimed discovery that the site became known. This kind of story is very, very common in archaeology; it tends to have deeply colonial roots that we need to recognize as we re-tell stories of both our modern and ancient pasts.

Cited:

John Griffiths Pedley, 1990. Paestum: Greeks and Romans in Southern Italy. (Thames & Hudson)

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^(1) To be fair, there are a lot of sites like this. In Europe, aerial photography carried out during the course of WW2 allowed us to see many sites which had otherwise been missed or weren't visible from ground level, and the explosion of academic funding and positions in the 1960s-1970s in part contributed to a significant uptick in archaeological exploration and excavation.