  In this letter we discuss new soft theorems for the Goldstone boson
amplitudes with non-vanishing soft limits. The standard argument is that the
non-linearly realized shift symmetry leads to the vanishing of scattering
amplitudes in the soft limit, known as the Alder zero. This statement involves
certain assumptions of the absence of cubic vertices and the absence of linear
terms in the transformations of fields. For theories which fail to satisfy
these conditions, we derive a new soft theorem which involves certain linear
combinations of lower point amplitudes, generalizing the Adler zero statement.
We provide an explicit example of $SU(N)/SU(N-1)$ sigma model which was also
recently studied in the context of $U(1)$ fibrated models. The soft theorem can
be then used as an input into the modified soft recursion relations for the
reconstruction of all tree-level amplitudes.
