Hi Ninjames, The LD MKIII is not an adequate amp for the Hifiman HE-500. Reason being mainly that is too low powered (100 mW at 32 ohm).
Saying that, I am running the HE-500 on it
. I found the CV4015 tube works pretty well and I think it combines best with 6N6P-IR (max dynamics whilst good stage size). I think it sounds surprisingly good. Nevertheless, it may be a better option to invest straight into a new amp.
Ignoring full tube amps, the cheapest offers seem to be from Schiit. A similar price range to LD MKIII would be the Aasgard solid state amp (some details here: http://www.6moons.com/audioreviews/schiit/5.html), better suited seems the Schiit Lyr (some info: http://www.6moons.com/audioreviews/schiit/9.html), which is more or less double the price. However, it is a tube/SS-hybrid amp and you can use your LD driver tubes with it. The HE-500/Lyr combination is widely recommended on Head-fi. In order to max it, you could pay a little fortune. Recently, I read that the Cavalli Liquid Gold is amazing for the HE-500, which is $3750... Generally, above Lyr, there seems to be no general consent about the go-to amp, Choices are Violectric V200, suitable Audio-GD amps, Schiit Mjolnir, Darkvoice 337 (which is actually similar priced to the powerful LD MK VI+) etc.
I have heard HE-500/Mjolnir and it sounds very impressive and better that the Lyr (IMO) but form factor and only balanced out are not ideal for me. I have found the Auditor, which is not very powerful (~4 x LD MKIII) but seems to match the HE-500 and my preferred sound signature well (neutral, non-fatiguing, detailed, wide stage, good bass).
To address exactly your question, I could not recommend a tube amp in a price bracket just above the LD MKIII. If you look out for it, make sure the output impedance is sufficiently low (ideally 5 ohm max) and consider a hybrid amp =).
Very best