As I said in an earlier post, quite a few animals went extinct in the wild (Przewalski's horses, European bison, scimitar oryx) but still had enough of a captive population to propagate the species. All modern Przewalski's horses descend from only nine captive individuals, but there is now a population of over three hundred in the wild due to careful releases, and the same thing happened with European bison.
For just about any animal this last that survived up until at least the 1800s-early 1900s, just have enough of a captive population hang on in zoos and their numbers can be bred back through careful management and possibly even re-released back into the wild. This should save the quagga/thylacine/Syrian wild ass/whatever species or subspecies that went extinct during this era.
Even in the modern day (80s to 90s), a lot of finch species went extinct in Hawaii due to invasive plants out-competing their native food sources and invasive frogs eating a lot of their insects. More careful management or eradication of invasive species earlier in human history could prevent a lot of similar extinctions on the smaller islands.