I did a quick review in https://arxiv.org/abs/1511.00388v3
Some more physics-oriented ones:
Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and
Physics, Part I: Basics.
Y. Choquet-Bruhat and C. Dewitt-Morette, Analysis, Manifolds and Physics. Part II: 92
Applications.
Some more mathematical ones:
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume I. Interscience Tracts in Pure and Applied Mathematics.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume II. Interscience Tracts in Pure and Applied Mathematics.
Kartik Prabhu