Let me for concreteness use the same language of classical field theory that you used in your question. It seems like your question is at least partially about the definition of what we call spontaneous symmetry breaking. Is this applied only to the ground state, or does it make sense to apply the same reasoning to any other state?
In fact, we already have a name for the situation where a given state, that is a given value of the field $\phi$, does or does not change under a symmetry transformation: we say that $\phi$ is or is not invariant under the symmetry.
Spontaneous symmetry breaking is a more specific concept, and indeed it is usually formulated as the requirement of non-invariance of the ground state $\langle\phi\rangle$. While being more limiting then mere non-invariance of any chosen state, this property has striking consequences. It implies degeneracy of the ground state, or in other words the existence of flat directions of the potential, which in quantum field theory translates into the existence of gapless excitations: the Nambu-Goldstone bosons. Furthermore, it implies that excited states are classified by irreducible representations of the unbroken subgroup (not the whole symmetry group) and that the Nambu-Goldstone bosons as a rule interact weakly at low energies.
The above said, it sometimes makes perfect sense to apply the same definition to other states than the ground state. For instance, an equilibrium state with nonzero density of a given conserved charge (typically the particle number), which is usually obtained by minimizing the grand-canonical Hamiltonian, can be equally well described by a time-dependent solution of the classical equations of motion, following from the microscopic (canonical) Hamiltonian. By looking at fluctuations of such a time-dependent background, one can generalize the Goldstone theorem to many-body statistical systems. Interestingly, one thus finds that such fluctuations retain most of the attributes of Nambu-Goldstone bosons, except that they need not be gapless: see 1204.1570 and 1303.1527 for more details.