I've played a little more but haven't gotten too much further. I did come across some things you'll probably want to know about. I'll spoiler it so as to not ruin anything for folks.
I've just gotten double jump from Basalt Woods. I'll include screenshots of the areas I'm talking about so you can see.
In Basalt, I noticed there were a few random places whilst climbing trees that let me stand on the air. Screenshot here (http://oi47.tinypic.com/mv2ljp.jpg). Not sure there's anything you can do about it but I figured I'd mention it.
I've also discovered a few places where I can find void screens. Found one by double jumping straight up off the gate (http://oi50.tinypic.com/9va4nq.jpg). There is one above this screen (http://oi46.tinypic.com/2ziziaw.jpg) and the one immediately to the left. Again, reachable by double jumping straight up. Finally, whilst at the mayor's house (http://oi49.tinypic.com/117a5vt.jpg), I found a void screen by double jumping up and to the right. I'm not sure if there's on to the left because I couldn't reach it. So it might not matter. In addition, double jumping straight up from the mayor's house leads into the bottom of the little inside area you have to climb to reach the mayor's house. It results in wall-swimming.
Anyway, that's all I have for now.
I've played a little more but haven't gotten too much further. I did come across some things you'll probably want to know about. I'll spoiler it so as to not ruin anything for folks.
I've just gotten double jump from Basalt Woods. I'll include screenshots of the areas I'm talking about so you can see.
In Basalt, I noticed there were a few random places whilst climbing trees that let me stand on the air. Screenshot here (http://oi47.tinypic.com/mv2ljp.jpg). Not sure there's anything you can do about it but I figured I'd mention it.
I've also discovered a few places where I can find void screens. Found one by double jumping straight up off the gate (http://oi50.tinypic.com/9va4nq.jpg). There is one above this screen (http://oi46.tinypic.com/2ziziaw.jpg) and the one immediately to the left. Again, reachable by double jumping straight up. Finally, whilst at the mayor's house (http://oi49.tinypic.com/117a5vt.jpg), I found a void screen by double jumping up and to the right. I'm not sure if there's on to the left because I couldn't reach it. So it might not matter. In addition, double jumping straight up from the mayor's house leads into the bottom of the little inside area you have to climb to reach the mayor's house. It results in wall-swimming.
Anyway, that's all I have for now.
Ah yes, to explain:
For the first screen: You are able to stand there for specific reasons that I cannot reveal to you until you've hunted down one of the bigger game secrets. That is just an unavoidable side-effect, but it shouldn't be too much of a problem.
As for the others, that was an oversight on my part. I was trying to make it so that you couldn't go back that way after getting the double-jump powerup, but it seems it was possible. Thank you for pointing that out. I will take the appropriate action for the next beta.