When it comes to general covariance, it covers not only linear(-affine) transformations which make up the group $GL(n)$, but also non-linear transformations. A very basic example of it  for instance is the choice of spherical coordinates:

$$t = t'$$
$$x = r \sin\theta \cos\phi$$
$$y = r \sin\theta \sin\phi$$
$$z = r \cos \theta$$

Of course the invariant line element no longer looks like a Minkowski-metric, but this is the essence of general covariance, that it deals  with any kind of metric. 

$$ds^2 = dt'^2 -dr^2 -r^2(d\theta^2 + \sin^2(\theta) d\phi^2)$$

(Of course linear coordinate transformations which are not Poincare' can also already change the metric --- for instance $u=t+r$ and $v=t-r$).
The given example of course does not change the property of the given space being flat. 
Actually a flat space cannot be become curved by a general coordinate transformation.