I see what you are asking for, I had the same issue a few months go. The problem with most cheap DI boxes is that the impedance is really low, like around 220KOhms. Most DI's just split the signal at the source, like a Y cable, especially passive ones.
So just like your speakers, the impedance divides. So your amp is 1Mohm or 1000KOhm for easy math. 1000*220/(1000 + 220) = 180KOhms. Which is way way to low, hence suck tone.
If you get a good DI box with a real 1Mohm impedance, then the math is simpler and the impedance is 500K which is acceptable.
If you look at the math, the final impedance will always be a bit lower than the lowest impedance. So if you can get a DI box with say a 10MOhm impedance: (10*1)/(10 +1) = .9MOhm which is certainly high enough.
For something cheap and good with a 1MOhm impedance, check out the active DI from Audiopile.net. $50 and they also sound pretty darn good.
Another option is to run a buffer before the DI box (boss pedal, DIY one, etc.). I found that for the ultimate purists, the buffer will alter the tone a bit as well, for metal, often for the better.
But with my EWI/Audiopile DI, I haven't had a guy complain yet and the DI's are perfectly good for re-amping/editing.
To really push the purism, I actually built a DI box using a high-end Burr Brown op-Amp, high input impedance with a unity gain buffered out and a line level output so the guitar DI had no coloration from the mic pre or anything. No transformers in the signal path, high end audio grade capacitors, bypassed electrolytics, etc. Freakin audiophile dummyness all the way... hey it was only a few extra bucks in the grand scheme, why not...
The buffer still alters the tone a tiny bit, mostly from the lack of cable capacitance. If you use a 2ft cable, the effect is the same.
With the DI signal, I can hear a difference, sort of like the difference between opamps or something. Real subtle as in I am not totally sure I could do it blind. And definitely not after reamping.
Long story short, I am not convinced it is worth the effort. Get a good DI box with a high input impedance and be done with it. I think some of the Samson ones have a 1MOhm input as well if you want to go cheaper.
But I highly recommend the EWI/Audiopile ones. Very well constructed and sound great!