Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227. Theorem 3, printed p. 222: "Let $\zeta=(z_1+\cdots+z_n)/n$, where $z_1,\cdots,z_n$ are the zeros of $f(z)$. If $E$ is connected, then $C$ is contained in the circle $|z-\zeta|<2$", $C$ the lemniscate $|f(z)|=1$ and $E=\{|f(z)|<1\}$; this is the result behind the site's remark that the constant $2$ can be reached when the set is connected: one disc of radius $2$ covers $\{|f(z)|\le1\}$ when the open set $E$ is connected (the site's remark names the closed set, a weaker condition that Pommerenke 1961 covers in the proof of Theorem 10(b), p. 107); the accepted partial claim on Pommerenke 1959. The paper does not treat the general case of this page's question. Library home: Pommerenke 1959 and its Pommerenke 1959, Theorem 3 page.

Pommerenke, Ch., Einige Sätze über die Kapazität ebener Mengen. Math. Ann. 141 (1960), 143--152, DOI 10.1007/BF01360168. Satz 3 (p. 149): a closed set $E$ of positive capacity is covered by finitely many discs whose radii sum to less than $2.59\operatorname{cap}E$. It follows from Satz 2, under which finitely many curves of total length below $10.36\operatorname{cap}E$ enclose $E$. The introduction (p. 143) applies it to the set where a monic polynomial of degree $n$ has modulus at most $r^n$, whose capacity is $r$, as a sharpening of the Boutroux--Cartan lemma. The site's commentary credits the bound $2.59$ to Pommerenke under its key [Po61], which its reference record resolves to the 1961 paper below. That paper cites this one as its [12], for the length bound in its Theorem 7, and none of its numbered statements gives the covering bound.

Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; Theorem 10 and the containment of a connected $E$ in the disk of radius 2 about the centroid, printed pp. 106--107, the accepted partial claim on Pommerenke 1961; Theorem 2, p. 98; Theorem 6, p. 102. Library home: Pommerenke 1961 (result page Pommerenke 1961, Theorem 10).