LOL, I really wish posters would take the time to read the OP's posts. He did specifically say "no towers".
I see no good reason to do 3.1 at this time, let me tell you why. You are looking for better sound, you can achieve excellent sound with a 2.0 or 2.1 set up. You have a budget of $560 for speakers, I suggest you do a 2.0 at this time. Just two full range bookshelf speakers. At that price, you can get some excellent sounding bookshelf speakers in the new market, or if you are brave enough spectacular sounding used speakers. Audiogon is a good place to look for gently used high fidelity speakers on a budget.
Later, when the budget allows, you can add a subwoofer. If you living arrangement changes and you can afford to have a full surround while maintaining the neatness get surrounds and a center.
I personally think you will get far more enjoyment from buying a stereo set of speakers right now instead of trying to get 3.1 or even 2.1. A good subwoofer is going to cost almost your entire budget. I think you owe it to yourself to get some sweet sounding bookshelfs.