Okay, after a couple rehearsals, I can now say
for certain that the best thing I have found to tighten up lows (besides getting a different cab), especially in a band situation, is play at a proper volume (i.e. loud) and turn off the deep switch, which I think is meant mostly for lower volume or single-coils. I used to always have Deep on kind of by default, but then I was wondering why I always sounded muddy on rehearsal tapes. Having Deep off also gives single-note leads more bite and complexity, especially on the bass strings.
Turn the deep switch off and turn the Bass knob up a little, just to like 1.5 or 2, or GEQ as required.
The III isn't supposed to have a crushing modern-style anvil of bass-- it's from the late 80s. You can get goofy low end out of it but it really kind of ruins the voicing of the rest of the amp when you do that.
On the tube side, though, if you have the scratch and want a beefy low end,
these NOS Siemens EL34s are worth every single penny. SED =C=s sound better in Rectos, which really do need the bottom tightened up, but these Siemens tubes own in Mark IIIs. As well, they seem to be very durable, I have been playing the sh!t out of a pair for a good seven months now and the tone hasn't changed one bit.