Next upgrade to an entry-level system?

a few thoughts

If the space is small, and you find the music to be INVOLVING, I would not change anything. You might try plugging the rear ports, re-adjusting the sub, and see if you find differences in imaging you prefer.

You could search for a different separate Phono Stage as suggested, but always get return option as I have returned several highly thought of mid-priced SS phono stages until choosing the two tube MM Phono stages I enjoy (McIntosh mx110z and Luxman SQ-N150 MM/MC). I use a SUT to listen to LOMC via my mx110z's MM Phono.

IF you get a larger listening space, keep this as a second system and start over, 1st with large 3 way speakers, try to find efficient ones that you like, which lowers the power needs, which allows you to try tube amplification.

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I assume you mean a factory provided AT95 cartridge body upgraded with a Shibata Stylus which is a very good choice for that TT, I would use that or the ML Microline version.

changing cartridges can be complicated, because:

which AT-LP120 do you have? Early 120 models allow arm HEIGHT adjustment, later models ... 120X .. (with X in the model number) have fixed arm height, thus you need to be careful and keep the cartridge height very similar.

If you have used or buy used LP's, they will sound much better if you thoroughly clean them and play with your Shibata or a Microline stylus shape. 

I disagree with the suggestions of cartridges with Elliptical stylus.

I don't know what your listening room/environment is, but you might want to look into that, as the room generally has more effect on sound than any component. For instance, if it's all hard, reflective surfaces and the sound is bouncing around all over (and I know people who have it like that), you can spend all you want and nothing is going to sound very good. 

in my opinion, speakers are always the dominant influence on your system. So, my suggestion would be to start with the best speakers you can afford. Best meaning what sounds best to you, not to someone else. Its not a bad idea to consult forums or reviews to build a list of speakers to audition but take all that with a big grain of salt. Individual tastes differ, and you won’t find many reviews that are not influenced by advertising dollars, “accommodation pricing“ or some other conflict of interest. After you get a list, listen to many as you can and pick the one that sounds the best to you. Then you will have a good foundation that you can build around as you upgrade additional components as your budget permits

Good luck!

Echoing (sorry) Larsman's point, setup makes a huge difference. A good way to approach the task is with guidance. Jim Smith is the guru of audio expo setups. He installs and voices esoteric equipment playing in awful rooms, and his reputation for optimizing systems is stellar. His book is Get Better Sound, available as an ebook for $18.70 at the website of the same name. The book has 202 tips that cover the gamut. Art Dudley reviewed it favorably at www.stereophile.com/content/book-review-get-better-sound. It is likely the best use of your upgrade money.

I get no kickbacks, but my system is the envy of my musician friends.

A new phono stage is an improvement that would get lost in the resolving ability of the amp and speakers.  NOT recommended.  The Yamaha has a decent amount of power so I would consider the speakers the first upgrade, then the amp.  If you buy another, higher quality integrated it may come with an improved phono stage anyway.