In this correspondence, we correct an erroneous result on the achievability part of the Rényi common information with order $1+s\in(1,2]$ in [1]. The new achievability result (upper bound) of the Rényi common information no longer coincides with Wyner's common information. We also provide a new converse result (lower bound) in this correspondence for the Rényi common information with order $1+s\in(1,\infty]$. Numerical results show that for doubly symmetric binary sources, the new upper and lower bounds coincide for the order $1+s\in(1,2]$ and they are both strictly larger than Wyner's common information for this case.