Phono preamp suggestions

There is a PS Audio GCPH on AG now in your price range. Great phono pre amp with lots of flexibility. Not affiliated with seller.

I've got agree on ps audio gcph. I owned the lp3 long ago it's a very quiet and nice for what it is just a little light in presentation. I use the ps Audio with parts connection mods in another room and when I compare it to a my other tube one at 3 times the price it holds its own. It's very quiet if you keep volume knob between 2 and 3 o'clock. It's very smooth and tube like in my system, maybe it's the mods. Pulls out a lot of information and very full sounding with plenty of headroom, the volume knob is very nice bonus for fine tuning gain. I never use the remote. I'm using it with a Soundsmith 2.4mv cartridge, but I've used it with low out put mc with good luck just can't go to low without a step up.

Turntable VPI Aries Scout with JMW 9 tonearm.

@paulcreed 
I've had both stock and Parts Connection modded versions of the GCPH. I currently use a Pass Labs XP-15, but I can't let go of the modded GCPH. Actually, the stock one currently listed on AG is priced a little high. They can usually be found for between $500 and $600.   

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.