Yes, such a closed geodesic always exists. See Theorem 1.1 of this paper by Basmajian, Parlier, and Souto (which I found by searching under the term "density of closed geodesics on a hyperbolic surface").
Now, to be honest, what you want is simpler than what is proved in that paper. Namely, it is well known that the geodesic flow has a dense (non-closed) geodesic $\gamma : \mathbb{R} \to S$. Take a subsegment $\gamma | [-M,+M]$, close it off with a uniformly short segment, and straighten to get a closed geodesic $\gamma_M$. For each $\epsilon>0$, if $M$ is sufficiently large then $\gamma_M$ is $\epsilon$-dense.