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I usually think the next step up from the 24AW is the Helleberg (usually Conn) 120s.

It's not just the mouthpiece. When I went back to tuba playing on a regular basis after being away from it for @ 20 years, the first thing I noticed besides needing to rebuild my flexibility and stamina was that low notes were thin and sharp, and this was with a Wick 1 on the Besson, one of the deepest funnels out there.

Drop the back of the tongue, think "oh," keep the embouchure firm, put a little more "lip" into the mouthpiece, relax the jaw a little bit down, and keep the air steady.

I agree with bloke on the unsuitability of the 24AW for most adult BBb players (TNFJ -- Don't go there -- I know there are some out there who do like it, and we're not going to talk about Eb -- but the majority of comeback players do not sound good on it). I did a demonstration once for a music teacher who was about to sell his BBb Reynolds Contempora. He had a 24AW, and it sounded thin. We were at a local music store, and I didn't have my mouthpieces with me, so I asked what was behind the counter. Among the choices was a Schilke Helleberg, which I said would do for my demonstration. Well, the improvement of tone was night and day. He got wide-eyed and said he didn't know the horn could do that. (Not surprising, as his primary instrument is trombone and he only played tuba occasionally). I believe he kept it.

Yes, by all means get a good 18, 18 copy, or 120S, but also work making your oral cavity feel like that you're trying to hold a raw egg in your mouth without cracking it for the lowest notes.

Bach 18 or Kelly 18, like bloke said. When I switched over to tuba from french horn, and when I picked up the tuba again 15 years later, that seemed to work well. (Old 3 valve banger in college, York Master 3/4BBb three years ago).

Good suggestion for affordable mouthpieces: go back into the TN archives and discover what was "hot" 3 years ago. Many of these will now be on sale, cheap.

Better: go back into the TN archives and discover what mouthpieces are used as benchmarks (to show that the new "hot" mouthpieces are "better".

Best: just play the g-d horn, using the mouthpiece it came with. It's not the mouthpiece, it's you.

Unless, of course, you are a collector...

Because I used to own this great horn I am in a rather unique position to answer. I used a PT88 at first, then a Wick 3L. Both did pretty well for me. I've also tried a Kellyberg with limited success and a Bach 18 with good success. But YMMV. I wouldn't deviate from the 24AW just yet. My recommendation - once you lock in with a teacher, work on it with him/her.