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The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γ _nlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.