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.