If you want something that will suit newborn to toddler, than you'll be best with an OSFA (one size fits all). I use Baby Beehinds Bamboo fitted. I have used them with Emily (my baby) since she was 2 weeks old, and my eldest dd has just toilet trained, and I had her in them 24/7.
The AIO's I've used (baby beehinds & itti bittis) don't require a cover.

Hope that helps some.