The next step would be a streamer and DAC. Sounds like you are talking a very low budget. Have a look at a BlueSound Node X. This should be a huge jump in performance.
The more you spend (assuming careful choosing) the better sound you will get up to the limit of your amplification / speakers. If I was in your situation I would go as far as a Schiit Yggdrasil (no I am not joking) DAC, and a used Aurrender N100 (or better yet N150) Streamer. That is my guess about where you would be enjoying the best sound quality from your system. Which should be far far better than what you get now. Using an iPhone as a source is pretty much the lowest level.