We started using a toddler pillow (Ikea one, which is very flat) when DC was about 2 years old and had by that point transitioned out of a crib. Over time, we switched to using a slightly puffier toddler pillow - don't recall the brand, just bought it off of Amazon.
Friends of ours swear by "pillow pets" for their toddlers -- kind of a cross between a stuffed toy and a pillow that kids snuggle up with in bed.