Like most things in life….It depends.
The degree or absence of DOMS seems to depend on muscle fiber type composition and athlete strength. This is from my observations of hundreds of athletes who have used this program. FT dominate athletes often get no soreness at all. Super ST athletes can be super sore for up to 3-4 days if they overdo it. Hence my warnings about not going full out for the first few workouts until you understand how it affects you. I do see that athletes have less trouble with DOMS when they have done cycles of to before. So, your experience this time with minimal soreness on the second time through an ME cycle is in line with most others see. If DOMS is severe, low muscle load activities are in order until you feel better. Running might be out of the question for a day or two.
In the end, the response is very individual. But the more you do this, the less painful it will be, and the long-term benefits are worth the short-term pain. I don’t think there is a way to alleviate DOMS.
I hope this helps,
Scott