Once more about free interpolation by functions analytic outside of a preseribed set

Let $\mathbb T=\{z\in\mathbb C:|z|=1\}$, $E=\operatorname{clos}E\subset\mathbb T$, $mE>0$. It is shown that (even if $E$ is nowhere dense in $\mathbb T$) there exist functions $f$ analytic in $\widehat{\mathbb C}\setminus E$ and satisfying some strong supplementary conditions (e.g. the uniform convergence of Maclaourin series in $\overline{\mathbb D}$, $\overline{\mathbb D}=\{z:|z|\le1\}$ and with boundary values of $f|(\widehat{\mathbb C}\setminus\mathbb D)$ of the form $\mathbb P_g$ with $g\in\mathbb C(\mathbb T)$, where $\mathbb P_-$ is the orthogonal projection from $L^2$ onto $H_-^2$). Moreover, some theorems about free interpolation by such functions are established.