Final year Uni student. Can I claim jobseekers allowance?

I'm a final year University student. I went to my local Jobcentre today to enquire about claiming JSA when I finish in May. The first guy told me I could claim there and then (although I know that's not true because I'm currently a full time student and full time students can't claim JSA!!). Another guy then came over and told me quite sternly I could not claim until September 5th. But after May I am no longer a student! I am fairly certain I have a job lined up for me in August/September, if not I am joining up to do a course in September, but I really will need some money from May until then incase I can't find anything in the meantime! From looking at other websites etc, it seems as though other final year students have been able to claim. Was the guy wrong, or can I really not claim JSA until September?