I've said it before and I'll say it again:
Take species that are alive right now, and re-introduce tem in areas where animals similar to those species became extint.
This is not unprecedented. In the pleistocene, there were horses in America, those became extint, and later re-introduced, with little or no effect in the ecosystem
Same with the Hippos in colombia (imported by no other than Pablo Escobar Gaviria, of "Narco" fame). Here, the efect on the ecosystem is low, but since the animals are very territorial, the populalition has a relationship with them of "Awe and respect"
In Venezuela there used to be an animal called Mixotoxodon Larensis, similar (but not related to) rhinos and hippos. It went all the way from brazil to Texas (the toxodon originated in patagonia, but our variation traveled more). We could re-introduce rhinos in venezuela and Brazil. Rhinos eat grass, like cows, so no biggie for the ecosystem, and are not a huge problem for humans (unlike Hippos hicha are VERY territorial).
In Venezuela we used to have a thinguie called the mastodons (other parts of america had them too, they came from the north), similar to elephants, so we may as well adopt elephants, either african, assian or both. Again, vegetarians, big, no biggie for the environmet.
Also, we used to have gavialoids (there are crocodiles, aligators, and gavials), but they became extinct, so may as well get gavials and "fake gavials" (which, funny enough, turned out to be true gavials ;-) ) which are on the brink to extintion, and re-deploy. Since they eat only fish, are no danger to humans, and could deploy in places with "bad" fish (think piranha or electric eels).
I think a similar case could be made about the other continents.
The opportunities are plentyfull, is just the disposition.