I am a Mayflower Society-recognized descendant of William White. You may already know, but the Society does have a 'William White Family' book detailing the descendants of his sons Resolved and Peregrine through 5 generations.
Yopu can also query Wikipedia and find new articles we created lasyt year for William White and his eldest son Resolved as well as enhancements we made for his other son Peregrine.
Additionally, you can query Edward Winslow in Wikipedia and find the sons names listed as his step-sons and about William's wife Susanna being the second wife of Winslow.
Also there is a recently formed William White Society - you may be able to find it on-line. If not, let me know.
Alton Rogers Desc. of Issac Allerton, Richard More & William White