Is it going to out perform the EagleTac they already have by a substantial amount? If not, then it will only be a disappointment.
I assume as you mention Eneloops, then all of your AA’s are these? Alkaline won’t offer the same performance.
The main issue you are facing is voltage. You need lots of AA’s to match a single or double Li-ion powered light. The same is true for amp draw, the NiMh’s just won’t match Li-ion. This doesn’t mean there aren’t options.
But I think you really need to clarify a few points.
When you say you need more light, is this because you want light up a wider area more fully. Or are you trying to light up something at longer distance?
As in, how exactly is the EagleTac not delivering what you want. And where you using it on charged Eneloops?
Next up, you need to give an idea of portability/size and budget.
The reason I say about what you want to light up, is because different configurations offer different ways of illuminating things.
For example, this is a single emitter light designed for more throw (XP-G2 C8):
While this is a 4x XM-L2 with 4x 18650 (SkyRay King):
The SkyRay is easily putting out over 4x the lumens as the C8. And it lights up the fore and mid ground very well. The C8 however puts light down on the ground in front of the electric poles and lights them up better (note the black line about halfway up the image for the SkyRay where it no longer puts light to the ground). And if there was something behind them, it would have illuminated that too, while the SkyRay wouldn’t.
What type of thing are you after?
The D40a is only 75 lumens more than the Eagletac he is using ( D40A 980 vs GX25a3 915 ) the D40A is quite floody but I doubt it will be any better than the Eagletac. The EAX Hammer is 2000 lumens from 8AA’s , over double the output. It is quite a bit bigger but not massive, it certainly wont fit in a pocket!
The Nitecore EAX is 8xAA and has 2000 lumens. If you want to stick to AA and want a much brighter light, it is the only way to go…
Something like th K40M would be better if you’re willing to use 18650.
Those free AA’s are probably alkalines so that’s not helping any.
I don’t think so. See:
Just out …or soon to come Sunwayman D80A no price or release date as of this posting.
8AA …2,000 lumens seen here Sunwayman D80A
How about the Sunwayman D80A.
2000 lumens form 8xAA
http://www.sunwayman.com/html/products/201410/162.html
EDIT: Sarratt just beat me to it! 
Hehe put some teflon around the threads to keep it tight. And shrink tube the head… You should be able to see your hands and the area surrounding your hands!