Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz 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?
Yes. This follows from the classical uniqueness theorem due to Osgood (the original paper appeared in 1898).
Osgood's Criterion. Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous on the interval $(0,a)$ and $\psi(u)$ is continuous on $\mathbb R_{+}$, $\psi(0)=0$, $\psi(u)>0$ for $u>0$, and $$\int_{0}^{\epsilon}\phi(t)dt<\infty,\qquad \int_{0}^{\epsilon}\frac{du}{\psi (u)}=\infty$$ for some $\epsilon>0$. Suppose that the mapping $f:[0,a]\times B_R(x_0)\to \mathbb R^d$ satisfies the condition $$\|f(t,x_1)-f(t,x_2)\|\leq\omega (t,\|x_1-x_2\|)$$ for any $t\in(0,a]$ and any $x_1,x_2\in B_R(x_0)$. Then the initial value problem $$\dot{x}=f(t,x),\qquad x(0)=x_0$$ has at most one solution on the interval $[0,\delta]$ with some $\delta>0$.
Osgood's theorem allows for the mappings $f(t,x)$ which are discontinuous at $t=0$. (Actually, the condition that $\phi(t)$ is continuous on $(0,a)$ can be replaced with an assumption of mere integrability.) Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that $f$ is continuous in $(t,x)$.
Moreover, Wintner showed that Osgood's uniqueness condition implies the convergence of successive Picard iterations to a local solution on a sufficiently small interval (A. Wintner, "On the Convergence of Successive Approximations", Amer. Journal of Math. Vol. 68 (1946), pp. 13-19).
