Uniform cover of a uniform space

Question

Let $X$ be a totally bounded uniform space with uniformity $\mathcal{U}$. For an entourage $E$ in $\mathcal{U}$, $\mathcal{F}=\{E[x] : x \in X\}$ is a covering of $X$. Is true that there is a uniform cover of $X$ which is a refinement of $\mathcal{F}$? Recall, a uniform space is totally bounded if every uniform cover has a finite subcover.