Next upgrade to an entry-level system?

I agree with other posters about upgrading the speakers next. There are quite a few well regarded speakers under 1K new. I followed a similar upgrade path and the new speakers helped reveal limitations with my other components. 
The next step would be the amp. Rotel, Cambridge Audio and Rega have well reviewed integrated amps under 1K. There’s real savings in the used market if you feel comfortable buying 2nd hand.

Speakers, and quality stands.  From your question and screen name, I take it you are a classical music lover. Klipsch speakers sound signature is not what I prefer for classical music. If I were in your position I would search out FBM, USAM, and here for a pair of used ADS L1290/2, L730, L780/2, or L880/2.  Make sure they have original drivers and .75” tweeters. All these models have ample midbass power and smooth extended highs.  Great clarity with no glare.  These are not dull New England AR type speakers…they are European inspired and were built to last.  The crossovers used mostly film caps and air core inductors. Rubber surrounds. Soft fabric domes with “sticky” damping compound. 
If any restoration is needed, Richard So does it best. 

AT about $1K, I would give a hard look at the Paradigm Premiere 200b, which often go on sale at around a bit over $800.  I have a pair driven by a Rega Brio in my second (bedroom) system, and they sound wonderful; my soul source for this system is streaming.  Excellent imaging for this price range, with a good balance across the frequency spectrum, although your sub will add some needed punch in the lower end.  I am quite pleased with them.  Worth a look and listen if you can, and they are manufactured in Canada.

@saintsaens_op61 IMO the best recommendation depends on your endpoint.  If you are looking to continually upgrade your system to a significantly higher level, then a major step-up in your speakers makes a lot of sense.  Just be aware that your source will really limit the benefits of the new speakers until you upgrade.  However, if you are looking for the largest immediate benefit to enjoy for a period of time I would do the phono pre and cartridge first.

Best of luck...

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.

..............................

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.