There are two ferries daily from Mytilene but the early one leaves at 6pm and the 2nd at 9 or 10 so its not possible to do this as a day trip. Our car rental company (Best) did not permit us to take the car. If we were with Hertz we could have left our car at the port and picked up another one in Chios but no doubt there would have been admin. charges. We decided to leave this for a future trip.

Not sure where you got that information, whilst the regular Mytilini >Chios >Piraeus ferries generally go in the evening, there are a few ferries at other times, it is possible to daytrip to Chios from Mytilini, sometimes SAOS, Agoudimos, Hellenic Seaways or NEL may have an early morning service, it is around 2 times per week.

I went about 08:00 to Karlovassi via Chios with Agoudimos ferries, the latter and SAOS ferries do change timings/days/routes month to month and year to year but it is usually possible.

The ferry agents on Mytilini harbour have the up to date info.
The return is more regular in the evening from Chios >Mytilini.

Thanks Ouzo, it was actually one of the ferry agents by the harbour that provided the information in early Sept. It appeared to confirm what I had found on the internet before leaving the UK. It looks like even the agents are baffled by the complexity of the Greek ferry timetables.