It is great having one person dedicated to continuity, but everyone should be aware of it. Simplicity is often the key to getting it right - if a character doesn't need complicated changes of clothes then why have them?
Problems often arise when you have similar scenes with one plot crucial difference; be extra vigilant if you are doing different scenes in the same location. eg. Where the breakfast table is set in bright daylight, then the table is unset and evening light fills the room. Make sure your shot list includes all the various differences (be aware generic shots may be unusable).
Take digital still photographs as reference. It's easy to miss details, and time-consuming to go through rushes to find them
Get people to change and leave their costume at the filming base, then you can be certain you always have the right costume to hand. Otherwise you may find someone starts a scene in a red T-shirt one day and finishes it in a blue jumper two days later. If this is not possible, make each person responsible for their own costumes and tell them to wear the same outfit each day.
When you are running short sessions and returning to film the same set again (particularly with animation) take care to ensure the lighting matches. Make a note of your camera settings, position of lights etc.