This article is a sequel to two earlier articles (one of them written jointly with R. Banuelos) on stability results for the Neumann eigenvalues and eigenfunctions of domains in $mathbb{R}^2$ with a snowflake type fractal boundary. In particular we want our results to be applicable to the Koch snowflake domain. In the two earlier papers we assumed that a domain $Omegasubseteqmathbb{R}^2$ which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of $Omega$ and those of its approximating subdomains satisfy a uniform bound for all sufficiently small t>0. The purpose of this paper is to extend the results in the two earlier papers to the situations where the approximating domains are not necessarily subdomains of $Omega$. We then apply our results to the Koch snowflake domain when it is approximated from outside by a decreasing sequence of polygons.