Let \(S(j)\) be a simple random walk in \({\mathbb{Z}}^ 2\), \(A\) be a finite subset of \({\mathbb{Z}}^ 2\), and \(H_ A(\cdot)\) be the hitting measure of the walk (from infinity). It is proved that \[ H_ A\{x: n^{-1}e^{- (\text{ln } n)^ \alpha}\leq H_ A(x)\leq n^{-1}e^{(\text{ln } n)^ \alpha}\}\geq 1-k(\text{ln } n)^ \beta \] for any \(\alpha\in ({1\over 2},1)\), \(\beta\in (0,{1\over 2})\) and any connected \(A\) of radius \(n\), where the constant \(k\) does not depend on \(A\) and \(n\).