Should I use an external Phono Preamp

That preamp has two Phono Inputs (either MM or MC) with very desirable extensive configurability. If it sounds great, done, find another way to scratch the upgrade itch.

"A total of 12 analog inputs¹ are located in the C12000 Preamplifier module: six balanced, four unbalanced, and two unbalanced phono inputs. The phono inputs are no longer set up as dedicated for Moving Coil or Moving Magnet cartridges; instead, both are configurable as either Moving Coil or Moving Magnet with adjustable Capacitance and Resistance Loading, along with adjustable Gain in 6dB steps from 40dB to 64dB. This provides maximum flexibility for you to configure the C12000 to best fit your turntable(s), cartridge(s), and vinyl record collection."

You have a removable headshell, any need for a Mono Cartridge? Or risk buying a used cartridge to get something unobtainable these days. I got lucky with 5 used cartridges, acquired 1 mono; 1 sapphire tube, and 3 beryllium tubes (no tube cantilevers in production these days). Also got lucky and found a very rare NOS Stylus (microwall beryllium tube, .0005" thick).

If you still want to try a different phono stage, I suggest a Tube unit, but keep in mind, anything Phono is 'preferred', not necessarily 'better'. I advise having a return option.

I used to have the same front end and cart with a different phono stage. 

you may want to consider a Sony HAT1 SUT with your Cadenza cart.and feed the SUT into your MM stage before trying something else. 

To outright beat the performance of the C12000 phono you are likely going to be looking at mid 5 figure phono stages such as a REF10 or TP 6.5 II or Boulder 2018.. 

I have a Cadenza Mono cartridge also

I have 3 of these, gold plated beryllium cantilevers, love their sound, and I can recommend the seller.

https://reverb.com/item/96890559-audio-technica-at160ml-mm-cartridge-in-excellent-condition

It includes an AT Headshell with azimuth adjustability.

Steve Leung, VAS Audio, very experienced rebuilder, just remarked to me how wonderful it is that these AT suspensions are still good after so many years.

I bought several things on Reverb, they support the buyer, did very well by me in the 2 instances of shipping damage.

The sonic one is looking for is only found by experiencing different Amplification of the sent signal. It is through experiencing different influences on an end sound produced that assists the user of the Analogue Source to discover the sonic traits that is most attractive to them. 

The more familiar that an individual becomes with the differences between end sound produced by different circuits and topology, will enable them to make educated choices to help them attain an Analogue Amplification that lends itself to ones unique preference for a produced end sound.

Firstly, My experiencing Aucurum Coil Wire is leading me to state that this wire type is capable of adding Richness of Tone, which is detectable in an end sound.

For myself when Aucurum Wire is used in the Analogue Signal Path, I prefer downstream support for the signal path to be auxiliaries to have a sonic trait that leans towards being transparent. 

If I am using a MM Input, the Step Up would be preferred using a Head Amp rather than using a SUT.

If the individual using a Cartridge with Aucurum Wire and is looking for a further injection of Richness of Tone. When the MM Input is being used, I would recommend the usage of a SUT. 

Either of the above are typically much cheaper than purchasing a Standalone Phonostage, and SUT's have for myself been both purchased and loaned items.

After my having used a reasonable range of SUT's, I am confident in my stating that differing models are capable of presenting different levels of noticeable tonal richness. The are SUT's I have not been in any way attracted to, more repelled from, due to the discernible tonal richness being detected in the end sound.

I have only had a small selection of Head Amp's used and these are always referred to as having discernible transparency.

For the OP, I have been to many demo's of Phonostages where comparisons between Phon's has been the reason for the demos being arranged. To date, I have not been a fanboy of Phon's demoed which are using Resistance Loading to interface the Gain required by a Cartridge.

For the OP, if Tonal Richness is wanted to be investigated, then an SUT and Head Amp may be all that is needed. Recently I added a link to a Post of a US Company offering home trial on their SUT.

If there is a want to improve the overall presentation, with improvement to micro dynamics, dynamics, detail, envelope. The search to discover the Phonostage capable of delivering this becomes much broader, I strongly recommend hearing any shortlisted in advance of committing, as synergy with the rest of the downstream auxiliaries is also critical to create a experience wanted to be maintained for a long time.