From Washington Services you are usually picked up at 5.30 am. I have travelled by coach on the last two holidays and each time it has been on a coach that started from Washington and went direct to Southampton with no more pick ups.
For years we drove to Southampton the day before and stayed at a hotel with car parking but after using the bus for the first time last November we couldn’t understand why we hadn’t done it before. It’s a doddle
Hope this helps you