I wanted to recommend Lee, but since you said it's too advanced... Well, to be fair, while his book is quite extensive, it is a very pedagogically written one too, so if you wish to study manifolds, at one point at least, you should read it.
I am not sure that's what you are looking for, but there are some GR books that discuss differential geometry in bit more detail and rigour than Carroll's book, these would be for example
- Wald: General Relativity
- Straumann: General Relativity With Applications To Astrophysics
- Hawking & Ellis: The Large-Scale Structure Of Spacetime
The last third of Straumann's book is essentially differential geometry, and he is quite rigorous.
For pure math books you could try
- Spivak: A Comprehensive Introduction To Differential Geometry
This is essentially a 5-volume grimoire, however it builds everything up quite slowly and pedagogically, and makes an attempt to build a bridge between the old formalism (indices, coordinates, etc.) and the modern one
- Isham: Modern Differential Geometry For Physicists
This one does not actually treat Riemannian geometry as far as I recall, but was written specifically for physics people, and also it has a nice account of principal bundles.
- Boothby: An Introduction To Differentiable Manifolds And Riemannian Geometry
About as advanced as Lee, I believe. Also this book does treat Riemannian geometry, as you can infer from the title.
This is a very advanced book that is quite hard to read, so I'd suggest visiting this later. However, it is also quite essential. Despite the fact that this (two-volume) book is quite old, it is still the standard reference in the field. The contents of volume 1 is what would interest you more, probably, as the most of Riemannian geometry is being treated there.