The simple rule is: There is no rule.
The Fugen-s in compositions of substantives has, apparently, developed from a genitive form of the composition (this seems to be the reason why an "-s" is used):
- des Gefechts Stärke --> Gefechtsstärke
- der Realität Verlust --> Realitätsverlust
- des Schnellzugs Wagen --> Schnellzugswagen (Swiss German), but Schnellzugwagen in Germany
- die Erklärung der Einkommensteuer --> Einkommensteuererklärung
- der Hochzeit Kleid --> Hochzeitskleid!!
As you can see from the 3rd and 4th examples, even this most obvious apparent "rule" is already broken by regional developments or administrative language. The fifth example is one that inserts an -s, even if this cannot have been caused by the genitive.
For your specific example Gefechtsstärke, Google finds a lot of (mainly) historical references that don't use the "Fugen-s", especially in official announcements of government and military. So, apparently, Gefechtstärke used to be a partially accepted form for some time, at least in the military.
The Fugen-s is inserted intuitively by native speakers according to the syllables that collide at the "Fuge" to make pronouncement a bit easier - for some combinations, an added "s" seems to be mandatory, for others, not (Zwiebelfisch and some other sources come up with a few examples that can be considered hints, but not rules). Even those combination hints are often not followed by common practice and with regional differences. Intuitive usage doesn't necessarily apply rules ...)
The German Rechtschreibrat has not bothered yet to come up with a definite rule (or, you could argue they evaded the topic because they couldn't find any rule). Apparently, you need to learn all those cases.
Answering to a comment: One of the assumed reasons for the Fugen-s is easier pronunciation and understanding. Compound words can come up with combinations of consonants that don't normally appear in German, like "td" (Institut[s]direktor), "tst" (Erbschaftssteuer) and would be hard to pronounce or combinations that are hard to hear, like "mm", "tk", "nnb"("Interimsmanager", "Hochzeitskleid", "Mannsbild").