My impression is that this is not a good follow-up to Rudin Chapters 1-8 in terms of sophistication.

Appropriate sequels would be:

\- Fleming. Functions of Several Variables

\- Loomis, Sternberg. Advanced Calculus

\- Duistermaat, Kolk. Multidimensional Real Analysis I, II

\- the later parts of "Analysis" by Amann and Escher

\- the later parts of "Mathematical Analysis" by Zorich

\- possibly "Advanced Calculus of Several Variables" by C.H. Edwards, though the level of sophistication is lower than the previous books (while higher than Hubbard)

\- Chapters 12-15 of "Mathematical Analysis" by Apostol or Chapters 7 and 8 of "A Second Course in Analysis" by Burkill; followed by any (rigorous) introductory book on manifolds. One possibility is "Differential Calculus" and "Differential Forms" by Cartan. Another is "Introduction to Manifolds" by Tu.

The assumed level of knowledge of linear algebra varies among these books.