The Tasmanian Tiger is one of my favorite extinct animals thought to now possibly be just cryptid. I think it has an excellent chance of still being around.
The Tasmanian Tiger is categorized as a "thylacine." The Tasmanian Tiger actually existed in Australia, the island of Tasmania and New Guinea. This was the largest marsupial (pouched creature).
It was believed to be extinct or nearly extinct on the continent of Australia by the time Europeans settled there. On the Island of Tasmania; however, it lasted much longer. Bounties on their dead bodies, as well as domestic dogs might have helped to obliterate this population.
The last thylacine died in 1936 (video below), officially declared extinct in 1986.
Even to this day, people often report seeing it. It's seen so often, that there has to be something to it. A breeding population would have learned to hide from man who was pursuing him and thylacine had bred extensively before the citizens began killing off their population to get rid of the "troublemakers."
In fact, a forestry worker in Tasmania saw one in broad daylight in 2011. He described a striped rear end, a long stiff tail and a weird rolling motion when it walked (source).
If a supposedly extinct cryptid does walk this earth, this is the one I would put all my money into. It is the most likely and the most witnessed, with a great deal of uninhabited land to occupy. In fact, there are plenty of videos on YouTube people have taken of potential thylacines. They're pretty badass and seriously, I hope the marsupials survived the slaughter because they are just magnificent wild creatures.
Here's one such film taken of potential thylacine:
When I saw this film of a supposed chupacabra taken by a police dashboard camera, I couldn't help thinking of thylacines. Strange, huh? It reminded me of the archived video (above).