Since the INT-60 has preamp outputs, I might suggest running those into the Apollon to try and get a better feel for the relative sonic contributions of the INT-60's preamp and power amp sections. I've never heard your Rockna DAC/pre or the INT-60, but some of the leaner character you report going from Rockna to Apollon may be due to having an additional preamp in the chain when running Rockna to INT-60. Or not, but it's relatively easy to test.
Regarding adding a tube preamp, I can heartily vouch for the ARC Reference 6SE, although it's $17k and I probably wouldn't have bought it solely for use with headphones (it's also the heart of my speaker rig). Smaller (and possibly previous generation) ARC preamps might fit the bill, but maybe not if you're looking for a lusher sound, as they tend to the analytical. Note that I use that term relative to many other tube preamps, not to solid state. I've personally never heard a solid state preamp that can reveal the harmonic textures of music like a great tube preamp can. I recently auditioned the Boulder 1110 hoping it might be the holy grail, but not even close to the ARC to my ears. Anyway, best of luck in your quest. Looking forward to further reports.