  We generalize soft theorems of the nonlinear sigma model beyond the
$\mathcal{O} (p^2)$ amplitudes and the coset of $\text{SU} (N) \times \text{SU}
(N) / \text{SU} (N) $. We first discuss the flavor ordering of the amplitudes
for the Nambu-Goldstone bosons of a general symmetry group representation, so
that we can reinterpret the known $\mathcal{O} (p^2)$ single soft theorem for
$\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $ in the context of a
general group representation. We then investigate the special case of the
fundamental representation of $\text{SO} (N)$, where a special flavor ordering
of the "pair basis" is available. We provide novel amplitude relations and a
Cachazo-He-Yuan formula for such a basis, and derive the corresponding single
soft theorem. Next, we extend the single soft theorem for a general group
representation to $\mathcal{O} (p^4)$, where for at least two specific choices
of the $\mathcal{O} (p^4)$ operators, the leading non-vanishing pieces can be
interpreted as new extended theory amplitudes involving bi-adjoint scalars, and
the corresponding soft factors are the same as at $\mathcal{O} (p^2)$. Finally,
we compute the general formula for the double soft theorem, valid to all
derivative orders, where the leading part in the soft momenta is fixed by the
$\mathcal{O}(p^2)$ Lagrangian, while any possible corrections to the subleading
part are determined by the $\mathcal{O}(p^4)$ Lagrangian alone. Higher order
terms in the derivative expansion do not contribute any new corrections to the
double soft theorem.
