As a 3rd year Applied Math major who owns several of these books I hope to offer some advice. Since you are pushing yourself, your time spent with Rosen would not be worth nearly as much once you start reading Knuth's Concrete Mathematics or Generatingfunctionology (credit to Chris Dugale in the question comments). Even if you don't have an eye toward coding/programming, you also would learn much about discrete math by perusing Knuth's The Art of Computer Programming, particularly Volume 1 (Fundamental Algorithms) or Volume 4A (Combinatorial Algorithms, Part 1).
I would prefer Andrews's Number Theory over Pinter's Abstract Algebra based on what's usually taught in high schools. But then again, if you have the patience and perseverance to read select sections of Gauss's Disquisitiones Arithmeticae, you'll gain great understanding in number theory topics too.
Though Schaum's Outline Series in Linear Algebra has good problems for practicing, its scope is not nearly wide enough to be a primary focus for your linear algebra skills. I used David Poole's Introduction to Linear Algebra for my school's linear algebra classes at RIT but I highly recommend Stephen H. Friedberg's Linear Algebra because of its chapters regarding Diagonalization, Canonical Forms, and Inner Product Spaces. These topics are powerful in themselves for linear algebra and probably not as valuable for contest math. However, learning about these topics will give you a solid foundation for important concepts later on.
What I'm trying to say is that Friedberg's discussion is a much more worthwhile investment of your time than Schaum's.
I don't about Coddington's Theory of Ordinary Differential Equations, but I can give an extremely highly recommendation for Nagle, Saff, and Snider's Fundamentals of Differential Equations and Boundary Value Problems. This text is an exceptional balance of theory, applications, examples, and exercises that gives a very clear and thorough exploration of ODEs and PDEs (Laplace's Equation, Heat Equation, Wave Equation, etc.).
I'm going to use Rudin's Principles of Mathematical Analysis in my upcoming semester for a Real Variables course. I imagine that this classic text would be worth some of your time for contest math but only if you have a strong handle on elementary single-/multi-variable calculus. James Stewart's calculus books are popular and decent for the most part; Ron Larson and Robert Hotsetler's calculus books seem to have clearer explanations and better exercises. [Avoid George Thomas's calculus books at all costs! They are terribly written.]
Though I can't confidently offer a "schedule" for contest math practice, all I can say in general is to bolster your linear algebra, calculus, discrete, and analysis skills by consulting the texts above.
Best Answer
There is a rather large list available here.
I would add to it (for really strong students)
This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.