  We provide $L^p \to L^q$ refinements on some Fourier restriction estimates
obtained using polynomial partitioning. Let $S\subset \mathbb{R}^3$ be a
compact $C^\infty$ surface with strictly positive second fundamental form. We
derive sharp $L^p(S) \to L^q(\mathbb{R}^3)$ estimates for the associated
Fourier extension operator for $q> 3.25$ and $q\geq 2p'$ from an estimate of
Guth that was used to obtain $L^\infty(S) \to L^q(\mathbb{R}^3)$ bounds for
$q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic
paraboloid in $\mathbb{R}^3$ based on the work of Cho and Lee. Finally, we give
some refinements for the truncated paraboloid in higher dimensions.
