  We establish several surjectivity theorems regarding the Galois groups of
small iterates of $\phi_c(x)=x^2+c$ for $c\in\mathbb{Q}$. To do this, we use
explicit techniques from the theory of rational points on curves, including the
method of Chabauty-Coleman and the Mordell-Weil sieve. For example, we succeed
in finding all rational points on a hyperelliptic curve of genus $7$, with rank
$5$ Jacobian, whose points parametrize quadratic polynomials with a "newly
small" Galois group at the fifth stage of iteration.
