I'm not familiar with the Moon LP3, but it looks like it's an entry level phono pre and probably the weak link in your analog setup. You probably need to budget $1500 - $2000 to "dramatically" improve sound quality, but in addition to some of the options mentioned, you might look for a used Musical Surroundings Nova Phonomena. You should be able to find one within your budget. Tons of loading options, a wide range of gain settings, and a very "black" background.
For $400 you could also have a Parks Audio Puffin. I think it's a great value and punches way above it's weight. I recently picked up a JVC QL-Y66F direct drive turntable from the 80's and mounted my old Shure MX97xE cartridge ($99) with a JICO SAS stylus ($200) and did some headphone listening last night to the 180 gm Led Zeppelin III remaster and... holy smokes!
The turntable, cart, stylus, and phono pre weigh in at just a little over $1K, but the sound quality was up there with some much more expensive vinyl setups I've heard. Mind you I was listening through a pretty high end headphone rig (Auralic Taurus + Audeze LCD-4), but the Puffin did not get in the way of the music in any way. It looks a little weird and it introduces digital into your analog chain which I know is anathema to some, but it's worth giving a listen to.
