Have you looked at tickets starting in DUB?
Direct DUB-SFO with Aer Lingus 9-15 Aug £831, WOW Air via Keflavik same dates £481.
Obviously you need to factor in cost of getting to/from DUB and the cost of an overnight hotel dpending on your risk appetite.
For June you can go Aer Lingus/Virgin Premium Economy via MAN on 6-13 June for £861, £464 if you go economy.