  The optimization of quantum control for physical qubits relies on accurate
noise characterization. Probing the spectral density $S(\omega)$ of
semi-classical phase noise using a spin interacting with a continuous-wave (CW)
resonant excitation field has recently gained attention. CW noise spectroscopy
protocols have been based on the generalized Bloch equations (GBE) or the
filter function formalism, assuming weak coupling to a Markovian bath. However,
this standard protocol can substantially underestimate $S(\omega)$ at low
frequencies when the CW pulse amplitude becomes comparable to $S(\omega)$.
Here, we derive the coherence decay function more generally by extending it to
higher orders in the noise strength and discarding the Markov approximation.
Numerical simulations show that this provides a more accurate description of
the spin dynamics compared to a simple exponential decay, especially on short
timescales. Exploiting these results, we devise a protocol that uses an
experiment at a single CW pulse amplitude to extend the spectral range over
which $S(\omega)$ can be reliably determined to $\omega=0$.
