Let $p \,{\geq}\, 3$ be a prime number, $F$ be a number field with $\zeta_p \notin F^{\times}$, and $K = F(\zeta_p)$. In a previous paper, the author proved, under some assumption on $p$ and $F$, that an unramified cyclic extension $N/F$ of degree $p$ has a normal integral basis if and only if the pushed-up extension $NK/K$ has a normal integral basis. This addendum shows that the assertion holds without the above-mentioned assumption.