Second Course on ODEs - Self Study - Book Recommendation
I will give you a recommendation of three texts on ordinary differential equations that I took from the "theorem-proof-proposed problems" approach.
However, I am sure that here at MathSE there are many experts who will be able to give you a better recommendation.
Barbu, V. (2016). Differential Equations. (First edition). Cham, Switzerland: Springer.
Hirsch, M. & Smale, S. & Devaney, R. (2013). Differential Equations, Dynamical Systems and an Introduction to Chaos. (3rd Edition). New York, USA: Academic Press.
Arnold, V. (2006). Ordinary Differential Equations. (First edition). Berlin, Germany: Springer.
I highly recommend the first text, since it is a book that deals with the study of differential equations in a very formal way. However, it requires the knowledge of solid concepts in real and complex analysis in addition to a previous course in differential equations with a solid background. It seems to me that if you have studied from Simmons' book of differential equations, you will have no trouble adding solid analytical knowledge to that.