I've always found southampton very helpful with zone transits, I tend to give them a call just after compton.
Occasionally they will give you a vrp to enter like Bishops waltham etc so its worth having these in your gps ready to go to if necessary.
If they're busy they will vector you around SAM and on occasions this has taken longer than avoiding their airspace alltogether and going east around the Isle of Wight.
LF