I'm not math-y enough to even begin to go about proving this, but just based on logic I believe you can just cut the circle in half to give you a semicircle with a diameter of $1$ unit.
This will fit the longest possible layout along the diameter, and I don't think it should be possible for the snake to lie in such a way as to break out of the semicircle.
If anyone can provide any kind of proof (or disproof), feel free to edit my post.
Update from Martin Frank
think of this picture:
now the baby snake (in red) is making an L (in three different angles for better examples) - you see you could cut some pieces on the side....