Narrative scenarios are all well and good but they are fundamentally an addition to an existing rules set and the best way to achieve a narrative is by baking it directly in. This is generally achieved by writing rules that are narrative, synergistic and emergent.
Part of the issue with writing "just another skirmish game" (JASG) and reinventing its various rules are that most of its rules consist of writing a skirmish game rather than simulating an event. If I write just another skirmish game there will be line of sight and a charge ability that will differ from simply walking into combat, not because they simulate the story I'm telling but because that's what goes into skirmish games. Take your game and another skirmish game of a similar scale but totally different theme (so if you've written a zombie apocalypse game find a game set in the arenas of Rome) then tick off each rule that exists in near identical terms in both games, these are, by necessity, not narrative rules. Rather they are just another set of skirmish rules. Remove any that do not directly simulate the events your game is about. This alone can create a surprising depth of narrative, by removing negative psychology effects your game of super hero combat makes more sense, a game of investigation lacking combat rules can say more about the story direction than any amount of actual rules. For remaining rules ask what effect they have on players and how that is appropriate to the story, most vitally in caution vs recklessness, clarity vs confusion and power vs frustration.
Caution vs Recklessness - Rules will tend to have risk/reward elements, rules with zero risk and potential reward will engender a reckless approach, a "why not" attitude. If your game is about punk street gangs, violence and death should have low risk and high reward to encourage a live fast die young attitude (this logic is part of why vehicles in Gaslands can re-spawn when destroyed). Conversely, if your game features characters not used or inclined to violence its engagement should be serious and extremely risky, or in our punk gang game non violent solutions should come with cost, players should approach caution with caution as a gang leader probably would view being seen to take the wimp's solution.
Clarity vs Confusion - People involved personally in even relatively everyday and civil disputes find the situation fairly confusing while those playing a war game experience almost god like levels of clarity. Obscuring information in a war game is insanely difficult to achieve, so much confusion is expressed by random events, but question which events are random to simulate confusion and which because of JASG convention. How many systems suggest confusion over what happens when an unarmored man is hit by a point blank shot gun blast? How many people lack total clarity as to what would actually happen? The use of clarity and inevitability when other systems use confusion and chance can significantly shift a game's narrative and feel from movie like to gritty or realistic.
Power vs Frustration - Players play games for a sense of power but they remain interesting when their goals are frustrated. As such players will gravitate to areas of power in games (not necessarily identical to moments of control), so ensure that what your game is about is what wins your game and what gives a sense of power to players. If your game is about mighty wizards it should not be won by thieves stealing treasure and its most powerful moments should not revolve around long shot crossbows. Rather, spells should devastate the table top and gain game winning mystic power.
It should be noted when counting the unique rules in your system those in the faction section, scenario area or campaign system really don't count. These are all exceptions from your rules, not exceptions from somebody else's. In addition try to avoid listing interesting or exceptional rules as a series of add-ons with brief narrative titles, they work much better baked into the game. If a character is meant to be an athlete it should be part of building that character rather than just buying them the "athlete" skill. If such abilities are simply bought in then they must create emergent game play narrative (see below), the narrative can never be dependent solely on non-game text. For the athlete skill if the rule is "gains 1 action point" if action points can be used to shoot machine guns then the meaning of that rule will never be "athlete" no matter how many paragraphs of flavour text you attach to it claiming that it is.
When two or more rules combine or chain together to produce a new result more powerful (or sometimes weaker) than each of the rules would create alone those rules are acting in a synergistic fashion. This is one of the simplest and most effective methods of creating narrative because this is how humans create narrative in day to day life. Thing one happens which causes thing two to happen, that's a story. Examples might be:
Sequential negative - When you take a wound you lose an action point, you need full action points to perform certain acts.
Sequential positive - If you pick up the hammer you get +1 to attack dice, on a 6 attack dice knock the target down.
Combination - For each friendly model in 1" +1 to the dice, for each enemy -1.
Practically any rule which gives a modifier will be working in combination with another rule, allowing players to combine a series of such rules into a story. "I charge full speed with the hammer at an enemy while two allies pin them down, kapow!". The vital point to ensure is that these rules cannot be combined to tell the wrong story. So in a card driven system an injury which removes cards should never be able to remove negative cards so that injuries can make characters stronger (unless you really need them to be chopping their legs off to stop the poison spreading), if you include charges avoid making walking into combat instead a better option, and so on.
Emergent rules and emergent narrative refer to situations which are available within an opening situation but not inherently built into it, rather they emerge from player action and interaction. In my game SSO (a board game rather than a war game but the point still stands) one of the missions requires crew to go outside a space ship to repair its solar panels, it is complete when all panels have crew in them. The emergent result is that players can intentionally shut down panels to make the mission easier, which fits with the narrative. Emergent rules are tougher to produce but they can feel highly rewarding for players as they provide a real sense of discovery. An example might be that allies give characters a bonus to rolls that is nonspecific and so can be used for combat but could be used to break down doors or climb walls. So the same rule can be allies distracting an enemy or giving you a leg up over a fence, in an emergent fashion, depending on the situation.
Ludo Narrative Dissonance
This is a fancy and fashionable term for what it feels like when rules and narrative don't match. So when injuries make characters stronger, mighty wizards fail basic spells or Samurai shoot their toes off, the play (ludo) doesn't match the story (narrative) which is annoying (dissonance). Generally this is the sort of thing that needs to be hammered out by sessions of playtesting but you can be lucky enough to catch it during writing. An interesting but less mentioned element here is that while mostly the break between play and story causes players to feel that the story is being lost it can result in a rule being perceived differently. So a rule with a violent name will be seen as violent even if its effect is multi purpose or potentially peaceful. This secondary effect can be beneficial to guiding the approach of your players, just ensure that it doesn't blind you to the reality of certain rules.
Long story short (and I realise this entry was long) narrative should never be laid on top of a rules set that does not provide a foundation for them. Narrative is vital in the building of a skirmish game worth writing and should be built in from the ground up. Look at every rule and step ask what story they tell, what story they could tell and what story they must be stopped from telling then allow the answers to guide your rule building and mechanics.