What's your purpose in wanting to "flush more water"?
Let us know what specific symptom you are trying to address.
Properly installed, the Drake has an excellent flush, amazing for the amount of water that it uses. I'm thinking this may be an installation problem more than anything else, or that something got stuck in your trapway (extra wax, tampon, GI Joe Doll, etc.). Did you buy it new? Did you install it yourself (i.e. so you know how it was done)? Is it level? Is your waste pipe clear and properly vented? If you hold the lever down just a second longer, does that totally solve your problem? If not, putting more water in the flush ain't gonna fix anything.
It's true that the Drake uses very little water. But it generally carts the waste off as well or better than even the 5-gallon monster I replaced with the Drake in my home. So before we start talking about tinkering with it, let's see if we can't make it work the way it is supposed to work for you.
Do you have the round bowl or the elongated bowl? You gave me the tank number not the toilet number. The tank is the same ST743E for the round toilet (CST743E) and the elongated (CST744E). Just curious what bowl we're dealing with. I also want to know: does the bowl say 1.28 or 1.6 gpf on it? (Toto now only makes one round and one elongated bowl for the Drake, the C743E and C744E; they no longer make the C743S or C744S, which was the bowl for the 1.6; so regardless of whether you have an E or an S tank, you should have an E bowl. I think it may say something like "1.6gpf or less" or something on it. I want to make sure you didn't somehow get an old S bowl with your E tank, although I think it would be pretty out-there if this was your issue; nevertheless, that anyone is unhappy with the Eco-Drake is odd to begin with.) So that's why we're here to help.
By the way, what you're asking to do is super-easy to do, and you don't even have to replace the fill valve. But even so, literally hundreds of discussions on here have never indicated it to be necessary. You will waste water, even more than 1.6gpf because the refill percentage on the fill valve is gonna be wrong. So before we go there, I want to see whether the problem isn't really something else, and my experience tells me that it likely is. (I know this because the owner of every poorly-designed non-Toto 1.6 gpf toilet, and of every improperly-installed toilet always wants to find a way to dump the whole tank into the bowl, and it never helps. Because the Drake is such a good product on 1.28 or 1.6, adjusting the water flow should be the last thing we look at doing. Indeed, every person we know on here who installed a 1.28gpf Drake in lieu of a 1.6 has said that they absolutely didn't need to try to increase the water flow because they were so happy with the 1.28. It happens over and over and over in this forum, so that's why I'm thinking your problem lies somewhere else and I'm asking all these questions. I look forward to hearing more.)
And just to give you some hope that we can fix your situation easily, perhaps without having to change the gpf, we had a poster on here not too long ago who was very unhappy with the flush on his Toto. That's a red flag, because he shouldn't be. IIRC, he either gave a great description or linked us a video. Right away, we said it isn't starting with enough water in the bowl. It is essential for the bowl to start with the right amount of water in it, because that's how it's designed to work. Sure enough, when we asked what was up with the refill tube, and whether it was clipped above and pointing into the overflow riser -- duh, no it wasn't. The guy connected it properly, flushed once, the toilet bowl refilled properly and thus started the next flush with enough water in it, and since has continued to perform like a champ. The owner was then happy as a clam.
Work with us, and I'm sure you will be too.
PS Here's roughly what the flush should look like:
Link to same toilet flushing paper: http://www.youtube.com/watch?v=rC_TP7mrVW0