  We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant
$L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional
hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The
analogous result had been previously established in dimensions $d=1,2$ using
the convolution structure of the inequality at the lower endpoint (an even
integer); we obtain the generalization by using tools from bilinear restriction
theory.
