Kid Icarus is part action, part adventure game. Worlds 1, 2, and 3 end with a fortress maze. The other stages are linear, but progress can still be saved with passwords. Below are my original passwords and maps.
1-4 Map (click for larger):
2-4 Map (click for larger):
3-4 Map (click for larger):
Similar to the New Game+ mode of Chrono Trigger, you can start Kid Icarus again after
beating it, retaining your statistics to make the game easier. And, also like Chrono Trigger, Kid Icarus has multiple endings (in this case, based on how many of your statistics are at their maximum level) which are easier to earn after beating the game.
Unlike some videogames (Castlevania III, for instance), enemies do not change in any way to make the game harder. In fact, since you are stronger, the game should only be easier.
For some reason, you cannot start 1-1 with the statistics you have earned from beating the game. 1-1 is always started with one strength arrow, one life block, etc. Passwords are only meaningful for the second loop from 1-2 on.