Uniqueness interval due to Picard–Lindelöf theorem

Question

So in my course, the interval $[t_0-\delta,t_0+\delta]$ where you know some IVP is unique due to Picard-Lindelöf is given by $\delta = \min\{a,1/L,b/M\}$, with $a$ and $b$ the (half)dimensions of some box $U \times V$ and $M$ the max of the derivative in this box. The IVP in question is $dx/dt = \cos(t^2+x), x(0)=0$, which is obviously Lipschitz with $L = 1$ for every $t$ and $x$. So $U := [-a,a]$ and $V := [-b,b]$ can be as large as you want to, which makes $L$ the only limiting factor for the interval. I understand why the interval is limited this way (it has to be a contraction to use the Banach fixed point theorem), but I don't understand why you can't just "iterate" this IVP by creating a new IVP with initial point $x(\delta-\epsilon)=x_1$ from the end of our unique solution, so you end up with a chain of unique solutions, making the IVP solution "effectively unique" on some arbitrarily large interval. Am I wrong?