Big improvement looks pretty nice 

I would also try see if the appearance is better when the bottom side has no outline.

All the advice above are very good in general in order to make the character more lively / real. If you'd rather keep it very surreal however, you may rather choose:
- a sinusoid curve instead of the bounce, like a Ghosts'n'Goblins spectre (https://youtu.be/uIhyjrXVhzg?t=53)
- no squash/stretch matching the movement, conveying is has no mass that would be subject to the law of physics
- a uniform light / not getting light from an outside source
An example of this is the Skel-Monsta from Bubble Bobble, that ignores gravity and walls, and it matches its design : https://bubblebobble.fandom.com/wiki/Skel-Monsta

edit: I had a try

i think you can make it more interesting by adding some secondary animations, like eye blinking and arms swinging

made you a fast example:


also squash and strech could help:

Dithering only really works if you can cover a large enough area with it. What you've done reads more like a few additional lines rather than dithering. That said, I think it is a step in the right direction, as it breaks up the artificial-looking horizontal lines, turning them into diagonals.
I think you're focusing a bit much on the sections of bark, I'd weaken them by breaking up the lines, and I'd instead create texture with rougher cluster edges.

(I also added some thin branches, as the tree looks unnaturally tidy to me without them.)

Thanks !

Since this message is old already, I actually have reworked it quiet a bit !

Heres how it looks like now

Idle animation 48 px height


walking animation 64px height



Hope you enjoy and if theres any tips.. Id be glad !