  The $L^2 \to L^p$ adjoint Fourier restriction inequality on the
$d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds
provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$,
if $d\geq2$. Quilodr\'{a}n recently found the values of the optimal constants
in the endpoint cases $(d,p)\in\{(2,4),(2,6),(3,4)\}$ and showed that the
inequality does not have extremizers in these cases. In this paper we answer
two questions posed by Quilodr\'{a}n, namely: (i) we find the explicit value of
the optimal constant in the endpoint case $(d,p) = (1,6)$ (the remaining
endpoint for which $p$ is an even integer) and show that there are no
extremizers in this case; and (ii) we establish the existence of extremizers in
all non-endpoint cases in dimensions $d \in \{1,2\}$. This completes the
qualitative description of this problem in low dimensions.
