Would the [Picard–Lindelöf theorem](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem#Picard.E2.80.93Lindel.C3.B6f_theorem) still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be [almost Lipschitz](https://en.wikipedia.org/wiki/Modulus_of_continuity) in y?

If not, are there any moduli of uniform continuity weaker than Lipschitz continuity that it is known suffice, or results indicating that there can't be any?