The correct approach might depend on what your desired outcome is. Will Slim eventually handle more than just the API, or will it just be for the API? If just the API, you could put an
index.php for Slim in a
api subdirectory and keep the source files out of
public. http://example.com/api/ An alternative might be to build it as a subdomain http://api.example.com/
If going the subdirectory route, depending on how your other app is setup you would need to adjust the index.php from the Skeleton to point to things another directory up.
// require __DIR__ . '/../vendor/autoload.php';
require __DIR__ . '/../../vendor/autoload.php';
Depending which webserver you are using you may need to make a few other tweaks. I recall people using Nginx needed to make a few other tweaks to get Slim working in a subdirectory.