The 49-year-old cases previous Crewe mentor Barry Bennell, a sentenced pedophile, manhandled him.
Cheshire Police said 11 individuals had approached since Woodward opened up to the world.
The Professional Footballers' Association (PFA) anticipates that that number will rise.
Tune in: 'nothing unexpected if manhandle as yet going on' - Pat Nevin
The four players to talk freely have each deferred their entitlement to obscurity as sex manhandle casualties.
In an announcement, White said he was sexually mishandled by Bennell in the late 1970s and mid 1980s, while playing for Whitehill FC junior group in Manchester.
He stated: "For various reasons, and for about two decades, I kept my experience mystery from my family and companions.
"While I accept all through my football vocation I have dealt with what had happened, I now understand the impacts of Bennell's activities were a great deal more broad than I knew then.
"A year ago I settled on the choice to accumulate my contemplations, ponder my experience, and recount my biography in my own words. I didn't set out to compose a tale about the mishandle, yet knew I would need to incorporate it.
"In doing as such I have grappled with the way that Bennell's activities affected practically every occasion and relationship in my life."
