I did a bit of a literature search and couldn't find much.

Let $\Sigma$ be a closed surface, and $PT\Sigma$ the projective tangent bundle (double covered by the unit tangent bundle $X$). Lenzhen and
Souto show in Theorem 1.3 that there exists a closed subset $\mathbb{X}(\Sigma)\subset PT\Sigma$ invariant
under the geodesic flow such that $1< dim_{\mathcal{H}} \mathbb{X}(\Sigma)<3$ (at least for *certain* hyperbolic surfaces $\Sigma$ with small systole).

However, the set $\mathbb{X}(\Sigma)$ is not the closure of a geodesic in $\mathbb{X}(\Sigma)$. This follows from the description (2.1), where the geodesics in $\mathbb{X}(\Sigma)$ have ends which are asymptotic to laminations (see Figure 1). So the closure of each geodesic will just be itself together with the lamination it limits to.

In section 5 of the paper, they describe invariant subsets $X^\tau \subset PT\Sigma, \tau \in [0,\infty]$, and show that $dim_{\mathcal{H}} X^0=1, dim_{\mathcal{H}} X^\infty = 3$. They ask if the function $\tau \to dim_{\mathcal{H}} X^\tau$ is continuous? If the answer is yes, and if $X^\tau$ is the closure of a single geodesic for $\tau >0$, then it would imply a positive answer to your question (at least in an interval $[3-\epsilon,3]$).