I don't know what the taxi situation is at 08:00 - the rank is often empty at 09:00 on the occasional mornings that I cycle from Swanwick station. There was absolutely no problem at 11:00 last Wednesday (not that that's any use to early travellers). I've never had a problem at Fareham.
You could try booking a taxi to meet you at Swanwick from one of the Fareham operators (I'm pretty sure that Swanwick Station is in Fareham Borough).