Ok, I dont mean Bigfoot or Nessie.
I mean creatures we know are real, but thought extinct.
Two such creatures there is some small hope for are the Ivory Billed Woodpecker and the Thylacine. Some sightings have been reported.
If you were a respected ornithologist and went out searching for that woodpecker, people wouldnt think your were a pseudoscientist or anything. Maybe a “Glass half full” overly optimistic, but not crazy.
So what other larger “extinct” creatures are still a maybe?
You can refer to the IUCN (International Union for the Conservation of Nature ) Red List, which documents the conservation status of different species across the globe.
Two categories - Extinct - and - Extinct in wild populations - are relevant. The next levels are Critical, Endangered and Vulnerable.
Thylacine - on the Extinct list.
Ivory-billed Woodpecker - this is listed as Critical, so it can be dangerously low, but not completely gone. You may have more upto date info than the Red List.
“Lazarus” taxa – thought to be extinct but later rediscovered – might also be of interest:
Good episode of Skeptoid by Brian Dunning on the Thylacine and whether it’s realistically still around, spoiler: virtually no chance. There have been massive searches and million dollar rewards to find some evidence and no trace has ever been found. He notes they look very similar to skinny dogs, so that seems to be the source of a lot of the “sightings”. He also touches on why they went extinct as well as other Lazarus Taxa.
Also - he has this episode on “Cryptids that turned out to be real”:
Both are well researched & interesting commentaries.
Regarding the OP:- if you were a “well respected” biologist and hunted for a thylacine now, you’d lose pretty much all that respect and be rightfully ridiculed.
For an ornithologist it may be similar, but to not be ridiculed you’d have to provide some justification as to why you suddenly felt the need to hunt for a bird that is considered virtually extinct. What new evidence is your hypothesis based on? What new technique will you can use to help find them or confirm none are there (i.e.: Sound, video, drones etc)?
In Australia, we have a bird called the Night Parrot (yeah, cool name to start with). Long thought extinct. First identified by western naturalists in the early 1800s, but no records after about 1900. Lives in the very sparsely populated outback and so unlikely to be seen ‘by accident’. Also, very similar in appearance to a related Parrot called the Ground Parrot, so any ‘possibly’ sightings had to be strongly authenticated, or they were dismissed as ‘probably ground parrot’. 2 dead specimens have been found. Suspicious calls were recorded - and argued over endlessly. It has always been the holy grail for Australian bird watchers - possibly not extinct - but totally unseen for over a century.
Anyway, in 2013, photos and videos surfaced, taken by a well-known Australian naturalist (who, however, had some history with possibly faking some earlier observations of other rare animals). Nevertheless, the sightings were eventually accepted. Other naturalists also investigated, and finally, live specimens were captured, recorded and DNA samples taken and confirmed. So - the Night Parrot has been re-discovered.
Interesstingly, although there is now no doubt about the Night Parrot’s existence, the original 2013 findings have been re-evaluated - some have definitely been found to be faked or manipulated. But at the time, they sparked new interest in the bird, which led to its re-discovery, so - were they bad? Just a great ruse to get more eyes looking in the right sort of spots? Pure fluke?
It’s a long saga in Australian bird-watching.
Nice one!
The Ivory-billed Woodpecker is 100% extinct, and the supposed re-discovery a few years ago was either wishful thinking or a hoax. They weren’t secretive birds and there haven’t been any confirmed sightings in at least 50 years.
Pessimist. You are probably correct, but still…
The 2004 sighting in Arkansas certainly wasn’t faked though it was pretty wishful thinking. I do have the T-shirt though!
Other more recent sightings probably fall into the same category.
100% is a bit much. 99% I can agree with. 
Sadly, the list of possible cryptids is only going to grow in the foreseeable future. Thing like Northern Right whales will be there before too long.
Folks came out in droves looking for them, but as far as I know, we only have a fleeting video that people still argue is a pileated woodpecker.
I’m no naturalist, but a possible candidate could be the South China tiger - no sightings in the wild for 40+ years, but not confirmed to be extinct in the wild (though I don’t know what “confirmed” might mean in this context).
Knysna elephants? Current views run from none, to one mature female, to a handful.
Dey Loy’s Ape isn’t too farfetched.
Good one.
No, it’s not too farfetched but doubtful. My opinion on that is that either it was a fake (but I dont think so) or a mutant spider monkey.
I don’t think De Loy’s ape would fall into the OP’s category of cryptids we know to have existed…
Aside from DeLoy’s “story” which was first told to a British newspaper, likely for personal publicity there have been zero other verifiable sightings or evidence, ever. Plus there are a lot of inconsistencies in his story versus known facts about his oil exploration trip. Lastly we have a verified eyewitness who says the entire story was a practical joke.
This should be in the category of joke cryptids like the jack-a-lope, but even with that one, more people claim to have seen it (at least with tongue in cheek).
Aquatic mammals are always hard to be absolutely certain about extinction. I have hopes for the Chinese river dolphin and the Caribbean monk seal. While both are very likely extinct, exhaustively searching all the tributaries of the Yangtze is difficult, and while the seal must come ashore to nurse, there’s many tiny islets that escape attention.
Nice. I guess we cant hope for a great Auk or a Stellers Sea Cow, eh?