I would go off the schedule for Friday the 11th -- there really is no reason (despite it being off season) as to why there should suddenly be no trains from Florence to La Spezia, I think it is a problem with the trenitalia site not having the correct times uploaded yet or updated on their system.
So while you might get to La Spezia and then on to CT, do keep in mind that it is off season for CT and many places might be closed. Not everything, as locals do live there year-round so I think you will get to enjoy it in a special way being there in December as opposed to June ;-).