The 18-year-old was initially on loan at Roots Hall for a month, and boss Phil Brown said: “Cauley’s has performed sufficiently well to warrant an extension to his contract.
“This first month he’s had has flown by because transfer deadline day was on a Monday and we had to get him in before then.
“I think Cauley’s a player of real promise and if he gets a start tonight or comes off the bench and gets a goal then his career will really be up and running.”
Woodrow has made two starts and to substitute appearances while with the Shrimpers but has yet to hit the back of the net in that time.