I've yet to be able to reverse engineer that high-warp table, but I do have a theory. The famous "S= W^3 C" has other variables such as local gavitational pull from stars, so that table is a good average within our galaxy. As Warp factor increases, the exponent on these other variables also increases, so that's how you get speeds of 22,000 vs. 343 for Warp 7.
In the background info for Star Fleet Battles, any ship with at least one working Impulse engine can use "non-tactical warp" to limp home at Warp 5.5 -- which also explains how pre-warp Romulans controlled their empire.
Tactical warp was invented later (which begs the question "what did they call non-tactical warp before tactical warp came along?") but tops out just over Warp 3, or in SFB terms, 30 hexes per turn plus one from an impulse drive for a max of 31 hexes. Interesting enough, the cube-root of 31 is just a tiny bit under pi. My theory is that tactical warp uses a spherical warp bubble that expands proportionally to the warp factor, and if it goes above a certain size it "pops". That limit is Warp = pi. The extra variables found in high-warp physics are so close to "one" (or "zero") that they don't matter -- they only become significant when warp is greater than pi.
To go faster than that, into the high-warp numbers above 3.14159, the navigational deflector dish distorts the warp bubble, extending the leading edge out millions or even tens of millions of kilometers, allowing the ship further to "fall forward" in the bubble to achive much higher FTL speeds. This also pushs smaller hazards, like comets and asteroids, out of the way and bends the ship's path safely around larger objects like planets and stars. However, if it runs into another ship's warp field, the bubbles loses their shape and forces the ships to "drop out of high warp" down to tactical warp speeds (or sub-light speed). It takes time and energy to rebuild the warp bubble, and you have to be far enough away from the other ship to avoid interference, hence the reason you can be forced into a battle in wide-open space. This is how ships "pin" enemy ships in F&E.
The above is just my theory, and ADB may have a different explanation how it all works.