What kind of budget do you have? new, used??? Modern, vintage?
I love the vintage sound for nostalgias sake, and keep several vintage set ups just because.....but my main rig is all modern, and I think with the exceptional clarity that your H III's have a modern design would suit your needs....
Rogue Audio has a wonderful tube integrated for less than 2K, that sounds like it costs 10K.....
Cayin has some very good sounding products that are reasonable...
Mcintosh has several newer integrated amps and they are a great deal considering they hold there value over time
If you like vintage a HH Scott rebuild, or Sansui, could be right up your ally
It's kind of like ice cream....lots of flavors to choose from, you just got to get out there and start tasting......Sorry if this made things worse for you....
Johann Gambolputty de von Ausfern- schplenden- schlitter- crasscrenbon- fried- digger- dingle- dangle- dongle- dungle- burstein- von- knacker- thrasher- apple- banger- horowitz- ticolensic- grander- knotty- spelltinkle- grandlich- grumblemeyer- spelterwasser- kurstlich- himbleeisen- bahnwagen- gutenabend- bitte- ein- nürnburger- bratwustle- gerspurten- mitz- weimache- luber- hundsfut- gumberaber- shönedanker- kalbsfleisch- mittler- aucher von Hautkopft of Ulm