Entry Level Tube Integrated Amps

Nice.  Sounds to me like you have many options given those speakers and subs..  Now it would be good to know what kind of sound you are seeking.  What is it that your currents amps are not bringing to the table?  What kind of music do you enjoy and at what spl do you tend to listen?

Interesting entry level would be the Black Ice FX10 or the Musical Paradise MP-301 Mk3 (both are integrated amps).

The FX10 is PP EL84 coming in @ around 12 watts for $800, or so, and the MP-301 is single ended (various output tube choices) coming in @ around 6 watts for approx. $500.

Both are well reviewed/Google them.

Once you move on either would be super for a desk top or other 2nd system (or as a gift to someone else starting out).

DeKaY

The Technics SU-G700 (paired with the Heresy IV's + the REL T9x) that I currently have are a clean, neutral sound. It's a very nice sound, but I'd like to get a bigger, broader soundstage and a warmer, meaty, fat midrange.

Sounds like what you are looking for is definitely a tube setup.  With those speaker choices you are in good shape.  Lots of good info provided already.  I will chime in with my two tube experiences.  I had a Jolida 502 brc (entry-level for sure) with 6550 tubes pushing about 60 watts.  Loved the imaging, soundstage, and midrange texture. But, not enough oomph for my Snell speakers.  So, went up to a Rogue Atlas 100 watt amp using KT-120 tubes.  Much better dynamics and bass but not as good in the imaging, soundstage, and midrange dept IMO.  Still very enjoyable though.  I have heard el34 based systems that blew me away with their midrange and palpability so maybe that is where you should look.      

@scotthiga  That is what I would have guessed.  You are looking for more meat on the bones. The warmth and humanity of tubes!

In my limited experience, the bigger push pull tube amps brought me closer to solid state.  So I went from a 14 tube amp down to a 3 tube SET amp and then I understood what tubes were all about.  That is not to say there are not push-pull tube amps that can pull off a similar feat--I just don't know which ones to suggest.

I can say that my Erhard Ray provides exactly what you are looking for in terms of sound quality.  But it may not have enough power or features.  Driving a pair of Omega monitors, 4 wpc is plenty of power for me, especially after I added a pair of REL 5/tx subs. While I love my primary system, I have to say that this little tube based system is wonderful to listen to.

Highly suggest you consider used gear so you can experiment a bit without taking much of a financial hit.  If the first one doesn't quite deliver the goods, you can move on.  I would also bet that which amp you like best will hinge on which one of your speakers it is driving.