The spoler video explained what I was doing wrong. It wasn't getting stuck out of bounds or anything, it was that I wasn't slamming hard enough against the edges the make the patches of grass appear.
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Oh, good. Turns out...even I am wrong in the video. It's not about how hard you slam the wall...it's all about what tile PatchOfGrass is next to, not LittleFoot. It just so happens that when back up to slam again, I realign, and then PatchOfGrass is aligned to the correct tile.
It's been changed in the Demo on Their Temple, to use LittleFoot's position instead of PatchOfGrass...let me know if that feels better. (You can still run into walls, but not as much, to my mind.)
Thanks for the great feedback!