I'm not sure how much Southampton Council might be able to help as Southampton / Eastleigh Airport is 100% in Eastleigh Council's area and outside Southampton's.
We do have loads of local aviation archives (including Eastleigh Airport) at Solent Sky Musuem in Southampton, simply phone up on a weekday and have a chat with an archivist.
www.spitfireonline.co.uk
There are also a number of books written by local historians about Eastleigh Airport. I have a 80 page or so paperback full of photos somewhere....I'll dig it out and post the ISBN if I can find it.
Edit: Found it: Southampton/Eastleigh Airport by David Hatchard. Published 1990. ISBN 0 946184 45 3. 56 pages and many pics.
Cheers
Corsairoz.