No that one doesn't close at all actually. Sammy loves to stick her head out but she also sits quietly and does not try to jump out
Also have you tried a sling? I made a sling for Sam and we love it. She doesn't try to jump out and I don't think she could if she tried since it doesn't have a flat bottom. I PM'ed you about the one that zips completely which is what it sounds like you need right now!!