I've spent time with your H600 in my listening room. Congratulations, it's a great piece. At this level, everything in and out will make varying degrees of difference.
If you wish to change the sonic presentation this is my short list, ordered by degree of impact. 1- external DAC, 2- external streamer, 3 power conditioning & cabling.
I would audition an external DAC and see what that gets you. Holo May, Denifrips, and the new Hegel DAC should all play into your preferred sonic signature. I don't think trying the Lmin mini was reaching far enough to make a big difference. Once you get into the $5K and up streamers, they really, really all sound different from each other in a highly resolving system.
Lastly, it might be worth trying a different power cord into the Hegel. A really good high current cord might get you enough to be happy and this will be by far the least expensive way to get there. Good luck and cheers.