A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever $λ_2, \ldots, λ_n$ are non-positive real numbers with $1 + λ_2 + \ldots + λ_n \geqslant 1/2$, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely $(1, λ_2, \ldots, λ_n)$. We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.