I would start with trying to determine if Slim is being called and if Slim actually runs on your new server. You may have already tried these steps, but I can’t make that out from your description.
I would first try to determine wether Slim is being called or you are getting a 404 page from the web server. If you know Slim is being called and runs, I would try a very simple Slim application and see if that works to determine if it is a web server issue or something in the code. something like:
<?php
require 'vendor/autoload.php';
$app = new \Slim\App;
$app->get('/', function ($request, $response) {
$response->getBody()->write("Hello from Slim");
return $response;
});
$app->run();