We study the emergence of non-compact degrees of freedom in the low energy effective theory for a class of Z2-staggered six-vertex models. In the finite size spectrum of the vertex model this shows up through the appearance of a continuum of critical exponents. To analyze this part of the spectrum we derive a set of coupled nonlinear integral equations from the Bethe ansatz solution of the vertex model which allow to compute the energies of the system for a range of anisotropies and of the staggering parameter. The critical theory is found to be independent of the staggering. Its spectrum and density of states coincide with the SL(2,R)/U(1) Euclidean black hole conformal field theory which has been identified previously in the continuum limit of the vertex model for a particular 'self-dual' choice of the staggering. We also study the asymptotic behavior of subleading corrections to the finite size scaling and discuss our findings in the context of the conformal field theory.