It depends what you mean by real symmetric perturbation. Let me assume that $A$ is a symmetric stochastic $n\times n $ matrix with first eigenvalue $\lambda=1$ of multiplicity $1.$ Define $\hat{A}_{S,\epsilon}=A+\epsilon S,$ where $S$ is a real symmetric $n\times n$ matrix and $\epsilon>0.$ Then it is easy to obtain an unbounded example. Indeed
take $A=\begin{pmatrix} 1/2&1/2&0\\1/2&1/4&1/4\\0& 1/4 &3/4 \end{pmatrix}$ and $S=\begin{pmatrix} 1&1&0\\1&0&0\\0& 0 &1 \end{pmatrix}.$ Consider 
$\hat{A}_{S,\epsilon}=A+\epsilon S.$ You can prove in this case that $\hat{A}_{S,\epsilon}$ is invertible and $\hat{A}^{-1}_{S,\epsilon}1>0$ for $\epsilon>0.$

Therefore the answer is: $\epsilon$ can be unbounded under no extra hypotheses.