Is this a one-piece or a two-piece? Is it one with the 3" flush valve? (Give us the model number in the tank if you can't figure it out.)
If it's a two-piece Toto with a 3" flush valve, just get the Korky 5030BP Large Adjustable 3" Flush Valve kit. The height of the overflow riser (which is what I assume you are talking about) is twist-to-lock adjustable, and it comes with a Korky flapper on which the flush timing is adjustable.
http://korky.com/FlushValveKit5030BP.html Korky sent me one to play with a while back when it first came out, and it's a good product.
Korky makes parts used in the Toto factory, including fill valves, flush valves, flappers, etc. So this will fit a Toto 3" two-piece and be the right quality.
It won't, however, work on a one-piece.
That said, I'm curious about your whole post. If you have the THU001z, then you have an old-model power-gravity lowboy toilet. Those by necessity have a lower and wider water column than some others. They also already use a flapper (the blue one) that drains a lot from the tank. I'm not sure that your premise that you "need more water" is correct. You're asking us to help you devise a "more water" solution, without telling us what is the underlying problem that you're trying to rectify with "more water".
Maybe we can help you if you give us a little more color and a little more about the problem you're trying to solve.