The minimum latency is 3.66 ms, but depending on the angle to the satellites involved it may be adding significantly less than 680 miles to the journey. Aka if it’s 340 miles east and 340 miles up that’s 480 miles to the satellite, adding just 140 not 340 miles.
Outside of HFT, most networks are far from the shortest great circle routes between you and the other end, which further completes the issue.
You misunderstood. (Ignoring the earth being a sphere.)
A satellite directly overhead means you need to travel to that altitude. However, if it’s not overhead light is traveling the hypotenuse of a right triangle where X is the distance to a point underneath the satellite and Y is the altitude of the satellite. That distance is the square root( X^2 + y^2). From there you need to travel to a different base station.
Assuming an ideal path where the satellite is directly between two locations that are 680 miles apart, that adds up to 2 * ( sqrt(340^2 + 340^2) ) ~= 961.7 miles vs 340 + 340 miles, or an added 281.7 miles not 680. In other words 1.414x the distance rather than simply adding 680.
Clearly the earth is not a flat and your very unlikely to be in that situation, but assuming you can reach several satellites at the same time it is likely one of them will be roughly in the direction you want to go.
Outside of HFT, most networks are far from the shortest great circle routes between you and the other end, which further completes the issue.