https://stresscenter.ucsf.edu/measures/hair-cortisol
The authors of the study mentioned above also monitored salivary cortisol. Total daily salivary cortisol output did not differ between people with CFS and controls.
People with CFS did have a statistically lower 'cortisol awakening response' (4.2 nmol/L vs 6.3 mol/L for the controls).
There is a huge overlap in values - some people with CFS had higher values, some healthy controls had lower values. So, even though statistically significant, it is not a very impressive, consistent difference and explanations like an, on average, later usual waking time for people with CFS could easily explain it.