We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $Δ$ is bounded by $O(n Δ^{7/5}/\log^{1/5-o(1)}n)$ for any $Δ$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge \log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in any interval of width $λ_2/\log_Δ^{1-o(1)}n$ containing the second eigenvalue $λ_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the typical support of a closed random walk of length $2k$ in any connected graph, which in turn relies on new lower bounds for the entries of the Perron eigenvector of submatrices of the normalized adjacency matrix.