I have had a look & there is a bus from Wickham Bishops (504). There is a bus from Witham, but for some strange reason does not service SHSG
Quite a few of the contract buses don't go all the way to SHSG. Or sometimes they do, but don't advertise the fact! I think it's because it's so much further than the others, and not many people from out our way put SHSG on the form. I imagine that the OP could change at Westcliff schools, and either join one of the contract buses that goes on to SHSg or get an ordinary bus from there.
Unfortunately, your DD would most likely be quite late for school if she tried to travel onwards from Southend Boys by public bus. There is no direct route. The best way,I think, would be to use the train to Southend Victoria then get the bus onwards to SHSG - you really want the train arriving at Southend Vic at 7.52 to be sure of being on time.
There are contract buses to the girls' school (one of DD's friends gets one from Chelmsford), but they don't always get there on time.