How should I upgrade? A new preamp with a phono stage or just a phono preamp?

Tubes/tubes/tubes, my preference

I always recommend FEATURES, the vintage tube preamps have it all. (except remote)

Using my McIntosh mx110z as an example only because I know it best (although I had similar full features on prior McIntosh SS C28 preamp).

http://www.berners.ch/McIntosh/Downloads/MX110_own.pdf

no MC input, but two switchable MM phono inputs which is handy. 

Has your needed tape loop; many inputs; trim controls to match volume of inputs when changing sources; Balance; tone; filters; loudness for very low level listening; and importantly the MODE switch.

Other makers, models exist with full features. This vintage stuff is what they made their reputations with!

I bought mine, then sent it to Audio Classics for test/replace parts matching existing (i.e. not 'upgrades', keep original). Best thing they did was put new gold plated rca jacks, the old ones were corroded, and they were slightly undersized.

Came back totally quiet, I couldn't be happier.

I paid around $1,100. and another $1,100 to Audio Classics (early 2019, prices for everything way up since then).

I get remote volume from my Cayin tube amp. I used to have remote balance, I think I will put my Chase RLC-1 back in play, perhaps only use it via tape loop. I have R2R, but only use mx110z Play IN (Tape monitor)

https://www.bonanza.com/booths/lbodin/items/1424839694?awc=6238_1681059762_53d236aa55c29a3c9c05558ec55fa1f7&utm_source=awin&utm_medium=cpa

It has two sets of identical outputs, how are you switching to speakers a to b now?

MUST have remote for RLC-1, no face controls, a door stop without the remote.

Mesch, why SS and not tube?

We came across an old Conrad Johnson tube phono stage and even in stock form it sounds really good.  If you want the best sound you can from LP then you would be compromised to go with an all in one unit.  We repair and modify components and even the old Counterpoint preamps that we upgrade are not as good as the CJ as a stand alone unit.  The B&K is a basic good preamp but overall really nothing special.  Same goes for the AI which is typically hard on tubes and as a preamp is bettered by the Prima Luna tube preamp linestage.  

Happy Listening.

My $0.02 if you want a tube phono, go separate, unless you listen to vinyl all the time, you'll be putting hours on tubes for no reason. 

Also my 2 cents, the modulus is OLD and likely easily bettered. It wouldn't hurt to verify that input and output impedance between the two are actually compatible. Good luck 

Would recommend a separate Phono preamp. My experience with integrated Phono pre's is that's where they cut corners. Plus if you change cartridge or TT at some point you don't need to mess with your system's. pre amp. 

Good luck!