Pick up and read Epictetus' Dialogues. (You could start with the Enchiridion, I keep a website with a lot of content on it at http://www.letsreadgreek.com/Epictetus/index.htm
). The Dialogues and Enchiridion (a summary of the Dialogues) is written in Literary Koine (in the NT, the book of Hebrews comes closest to Literary Koine); Epictetus, a Stoic philosopher of the 1st-2nd century A.D. shares a vocabulary which is eerily similar to the New Testament. The content is a practical discussion of philosophy (street level stuff), so there are plenty of stories and dialogue. Epictetus' dialogues were transcribed by Arrian, who also wrote the Anabasis of Alexander in Attic Greek -- There is a book on those differences also. There are also several books in German that also talk about Epictetus and the NT, and one English book. So there is a lot of content that can be compared to the NT and improve the range of vocabulary and structures of the Koine learner.
Reading Epictetus is better than reading the papyri, Jewish inspired works such as Joseph and Asenath, the semiticized Greek Apocryphal books (several were composed in good, non-Semitic Greek), Pseudepigrapha, Apocryphal Gospels, Didache, and other Christian/Jewish early literature. The Apostolic Fathers is another good read, (Holmes' volume is great, and has great introductions to each book, along with a translation of each book). Josephus would be another option, and there are some online resources for his writings. Josephus' Wars are better Greek than his Antiquities. The Greek church fathers are harder, as is Eusebius, ktl. They often wrote in a dialect closer to Attic. As with all ancient books, you really do need a commentary to make sense out of stuff once in a while.
Epictetus has a sort of special vocabulary for his philisophical terms that a reader needs to learn. (You can find a list on my site.) But I don't think a person could go wrong going down this road. And if you can read Latin or German, there are vocabulary lists (with extended glosses for the Enchiridion). There is a modern English language commentary on the Dialogues Book 1 - very helpful for bringing to light the historical context which you really need to know to understand the content. There are also several helpful works in French. Buth has a recording in Restored Koine audio of one of the dialogues. Seddon has a modern book on Epictetus' philosophy and terms. Boter (2007) has a new critical text of the Enchiridion. There is a Neo-Platonist commentary (modern edition) and a Christianizing version of some of the dialogues. So the content for this Literary Koine work is impressive. And you can find the resources you need to really learn Koine. Some professor of Greek is supposed to come out with a reader's guide of the Enchiridion (one of the Greek reader series that have been published lately. I cannot find the email I received from the publisher telling me about this person (it was not Steadman http://geoffreysteadman.com/
) -- perhaps I can search some more. That was 1 year ago, and I have heard nor seen nothing since).
And if you want to learn the difference between Koine Greek and "2nd century Restored Attic Greek," you can pick up the Anabasis of Alexander (and related book) and try to compare the two dialects. So there are a lot of possibilities and resources available to learn the Koine of Epictetus. And since you are into homonyms, antonyms, synonyms, ktl., it is just this sort of Greek you need to expose yourself to.