  It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff
dimension $\dim A \in (3/2,5/2)$, then for a.e. $\theta \in [0,2\pi)$ the
projection $\pi_{\theta}(A)$ of $A$ onto the 2-dimensional plane orthogonal to
$\frac{1}{\sqrt{2}}(\cos \theta, \sin \theta, 1)$ satisfies $\dim
\pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim
A+1}{3} \right\}$. This improves the bound of Oberlin and Oberlin, and of
Orponen and Venieri, for $\dim A \in (3/2,5/2)$. More generally, a weaker lower
bound is given for families of planes in $\mathbb{R}^3$ parametrised by curves
in $S^2$ with nonvanishing geodesic curvature.
