= Solution
Yes. Quotient only the first coordinate modulo $\mathbb Z$. The resulting process lives on $\mathbb T\times\mathbb R^2$ and the image of $S_1$ is the radius-$\epsilon$ ball about $(0,0,0)$. The two noncompact coordinates form <planar Brownian motion>, which is recurrent; during its infinitely many returns to a smaller disc, the independent circle coordinate has a fixed positive chance of lying in the required interval. The <Strong Markov property> then shows that the target ball is visited infinitely often. Lifting back proves that $S_1$ is hit at unbounded times.
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