Considering a PhD and/or a career in academia?
I recommend reading Katabasis by R.F. Kuang and Principle of Mathematical Analysis by Walter Rudin
Want to get a taste of first year PhD level econometrics?
Try the first 10 chapters in Bruce Hansen's Econometrics and Large Sample Estimation and Hypothesis Testing (Handbook Chapter) by Whitney Newey and Daniel McFadden
Interested in becoming an econometrician?
Here are some lecture videos, notes, and textbooks that I found rather helpful for my line of work.
Lecture videos on Probability Theory and Stochastic Processes by Todd Kemp
Lecture videos of PhD level Econometrics by Ivan Canay
Advanced Probability for Statisticians and Empirical Process Theory by Kengo Kato
High-Dimensional Probability by Roman Vershynin
U-Statistics: Theory and Practice by A. J. Lee
Quantile Regression by Roger Koenker
It is rare to feel completely mathematically prepared for research, as the new demands are often unpredictable. At a certain threshold of foundational knowledge (mostly real analysis and probability theory), rather than aimlessly studying more advanced mathematical topics, a more effective approach might be to begin working on a project and then acquire the necessary mathematical tools as the need arises.