You can't do any better than $7$. This follows from

Karoubi, Max; Weibel, Charles A., On the covering type of a space, Enseign. Math. (2) 62, No. 3-4, 457-474 (2016). ZBL1378.55002.

in particular Theorem 5.3 in the [arXiv version.][1]

The strict covering type of a space $X$ is the minimal cardinality of a good cover, denoted $\operatorname{sct}(X)$. This is not a homotopy invariant, so Karoubi and Weibel introduce the covering type, defined by $$ \operatorname{ct}(X)=\min\{\operatorname{sct}(X')\mid X'\simeq X\} $$ Obviously $\operatorname{ct}(X)\leq \operatorname{sct}(X)$. In Theorem 5.3 they use cohomological arguments to show that $\operatorname{ct}(T^2)=7$. [1]: https://arxiv.org/abs/1612.00532

You can't do any better than $7$. This follows from

Karoubi, Max; Weibel, Charles A., On the covering type of a space, Enseign. Math. (2) 62, No. 3-4, 457-474 (2016). ZBL1378.55002.

in particular Theorem 5.3 in the arXiv version.

The strict covering type of a space $X$ is the minimal cardinality of a good cover, denoted $\operatorname{sct}(X)$. This is not a homotopy invariant, so Karoubi and Weibel introduce the covering type, defined by $$ \operatorname{ct}(X)=\min\{\operatorname{sct}(X')\mid X'\simeq X\} $$ Obviously $\operatorname{ct}(X)\leq \operatorname{sct}(X)$. In Theorem 5.3 they use cohomological arguments to show that $\operatorname{ct}(T^2)=7$.