I felt that a lot of aspects of English grammar suddenly clicked for me when I started studying foreign languages. I had never had an English class that went much beyond the more familiar aspects (noun, verb, adjective, adverb); in contrast, foreign language classes jump right in to the specifics of verb conjugations, clauses, participles, gerunds, etc. For example, learning about the subjunctive tense in French finally revealed to me why we say “I wish I were” instead of “I wish I was”. The subjunctive is so almost-vestigial in English that it never seems to get much discussion in English classes.

Another reason that this is useful is that when you have only one system to look at, everything seems either arbitrary or obvious. When you have multiple systems to compare, you start to see what the different options really are (e.g., word order (consider English vs. Japanese), gender (so much more emphasized outside of English), case-markers (used in languages that aren’t strict about word order, like English)). It’s fascinating. Almost enough to inspire one to want to create one’s own language, once you see all of the different knobs you can turn!

So I’d say that another way to improve your English is to learn more non-English. :)