I like how you infuse the story into the game and I can definitely see the effort to put all the neon signs around town - so 5 stars for theme ;). Here is a detailed feedback for you as you'd asked for:
Perhaps to reduce the need to restart the level when the player get stuck, is to add an "undo" or "rewind" button? You can limit this to like 3 or something so it doesn't get abused :P This is because early in the beginning, I already fell below before I activated the platform that closed above me and didn't know how to "undo" the time travel and so I had to quit and start over because I just got stuck at the bottom.
Even though I agree that this game mechanic makes for very interesting gameplay, but perhaps need to give more leeway for such instances ;) Keep up the good work!