Best way to move a spider along a web?

I am planning on making a small 3D game where in the introduction you play as a spider on a web. I have not decided if I am going to use first or third person, but I will likely give the option to toggle. While on the web, I want the spider to move smoothly along the thread and only be able to in the direction of the thread. For example, the spider could go forward or do a 180, but not turn to be facing away from the thread. At intersections it should be able to change direction towards any available path.

I would like to keep this simple since it is such a small part of the overall game, but I am not sure how to approach it. How should I set up the web's model and colliders? How do I keep the movement smooth without getting caught on the colliders with such a narrow platform?