  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 IID 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 at NLO is 3/2, signaling a
departure from the leading order behaviour. Both pieces of information provide
clues for a non-trivial double-scaling limit, for which we put forward some
precise conjecture.
