First thing you need to do is work on mic positioning. Getting a new mic and placing it on the piano or near the hammer strike point isn't going to yield much improvement if any. Mic placement is more important than gear acquisition. Yes, there are inside the piano mics and mic techniques, but your budget would rule these out. Inside the piano mic placement is typically used in pop and jazz recordings.
Turn off auto gain, or whatever it is called on your device. For music recording it is best to set gain manually. Play some loud stuff on the piano. You want to set the gain so that there is no clipping when you are playing loud. Generally you would work with the loudest sounds you anticipate making in the recording.
Start with placing your mic/recorder/phone about 6 to 10 feet away from the piano, 4 to 8 feet up from the floor. Record a sample, listen, evaluate, take notes. Move the recorder around the room, rinse and repeat. You are looking for the sweet spot in your room. Alternatively, you can have someone else play the piano and you move your ears around the room as if they were microphones.
IMO, the portable audio recorders by makers such as Zoom, Tascam, and Roland offer the best compromise of affordability, ease of use, and audio quality. They will generally outlast the life of a smart phone and in some cases offer a bridge to acquiring higher end recording gear. Some models have XLR inputs to accept studio mics if you want to upgrade in the future.