The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately color uncolored edges $e \in B$ in their respective color and $P_1$ begins. Whoever completes a monochromatic copy of $G$ in their own color first, wins the game. If no one claims a monochromatic copy of $G$ in a finite number of moves, the game is declared a draw. In this paper, we give an infinite set of 3-uniform hypergraphs $\{G_t\}_{t \geq 3}$, such that $P_2$ has a drawing strategy in the Strong Ramsey game $\mathcal{R}(K_{\aleph_0}^{(3)}, G_t)$. This improves a result by David, Hartarsky and Tiba.