Wikipedia says the following on fight distance in relation to fuel consumption overall:
For long-haul flights, the airplane needs to carry additional fuel, leading to higher fuel consumption. Above a certain distance it becomes more fuel-efficient to make a halfway stop to refuel despite the energy losses in descent and climb. For example, a Boeing 777-300 reaches the tipping point at 3,000 nautical miles (5,600 km). It is more fuel-efficient to make a non-stop flight below this distance and to make a stop when covering a longer total distance.
I have heard more about this, and this seems to be the reason that ultra long-haul flights aren't as available as they could be and when they are available they are more expensive (because they are the fastest needing no extra time to stop).
How can a traveler use this knowledge to choose flights more efficiently? Let's say I don't care that much about a longer travel time and don't mind a stopover, how can I compare two flights in terms of fuel usage?
Let's say I'm comparing AMS-BKK and AMS-DOH-BKK (or LHR-SYD to make the issue even more relevant). The first is a direct flight and it's quite a long distance, so it might benefit from a stopover somewhere. Doha is a stopover that divides the route into two equal (almost) legs, so it might be more efficient. On the other hand, Doha isn't on the flight path AMS-BKK, meaning you'll have to travel some extra distance overall (based on the triangle inequality principle).
How can I, as a passenger with internet access and some knowledge (but not all the knowledge that airlines have), compare the fuel use of those two (or any other two) options?
There are of course other factors, for instance, different planes may have different fuel consumption characteristics so some may be more or less efficient on different ranges. Another factor might be that the number of people per flight counts, for example if you're flying alone in the plane then all fuel is spent for you, whereas in a full A380 passenger jet you 'share' the fuel burden among all passangers
Therefore, answers will probably have to use approximations and maybe some assumption, that's fine (though please indicate which approximations or assumptions are used). Answers could consist of (online) tools or rules of thumb (e.g. a comparison based only on the distances of the different legs).