Interesting!
To be quite honest, I never cottoned to the Jolida 502B, for exactly the reasons you listed, Sprinkbok10. I always felt the 202/302 amps were much better sounding. The 801 seems to correct what the 502 does wrong, but still has a touch of that 6550 hardness.
On the cheap end, you could have your amp converted to a 302, and use the EL34 tube, which provides the midrange and treble you seem to be searching for. You would get away at VERY low cost by going down this road, a simple reconfiguring of the amp, if I am correct.
Converting it to triode at the same time turns it into a TRUE giant killer, though you may want to try that first, and run it with KT88 tubes. I think I would investigate that, actually. The JJ KT88 is my favorite "normal" output tube, very sweet, with an enticing, punchy bass, and holographic highs.
On the sonic side, you will not find a tube integrated that compares with the Jadis. At least, that was the conclusion I came to in 1999 after basically trying everything, power amp or integrated - ARC, Cary, CJ, Jolida, Manley, Pathos, Quicksilver, Rogue, VAC, VTL. As far as most other tube amps go, they may be quite good, but believe me, a Jadis is a really special product.
The only product that I found on the same level is Air Tight, so that will comprise another recommendation I can give to you. Air Tight, the Tamura transformers! Like the Jadis, not deep, but on the wide side. In my opinion, Air Tight and Jadis are the Rolls Royce of tube amplifiers, and yet they are so underrepresented/underappreciated in our hobby today.
It is a true tragedy what has happened to the Jadis brand here in the USA over the past 7 or 8 years, but as they have a new distributor (Pierregabriel), I am holding out hope for things to change. Finding some dealers in the USA would return the marque to its former glory as seeing and hearing the product really leaves an astounding impression. I honestly had tears in my eyes when I played Dar Williams' "Southern California Wants To Be Western New York" during my JOR audition - and that was with Monitor Audio speakers no less! Never had THAT happen again to me, no matter what the system.
I'm not sure that any of what you will find can sound any better than running your Jolida, converted to KT88 triode operation.
GOOD LUCK!