What about skill improvement?
In Book 2: Starships of the Traveller Little Black Box, it talks a bit about getting education and training. Easy enough to Apocalypse it up a bit. Here’s a different way to say something similar, drawing inspiration from ( or ruthlessly pillaging, depending on your POV) Apocalypse World and Burned Over:
But, Judd, you ask, what about stats?
I like that we’re saying that the limits of the human body aren’t interesting to us. What you use to change the world is your training and in order to gain training you have to go out and interact with the world.
What about non-basic moves?
Non-basic moves are all tech. Maybe the True Sword, a networked blade with an embedded AI, gives you a move like the Gunlugger’s Not to be Fucked With, where you fight as a gang. Certain drugs give you access to psychic powers.