But, 12:01 WHERE? At the airport's hub time-zone? Or in my time-zone? I'm often at my computer at odd times so I will check out this theory.
I've checked flights to Europe on a couple of websites recently: travelocity, orbitz, sidestep, etc. There is sometimes a wide difference in price between airlines. I look closely at the connecting airports, since the budget flights almost always make at least one connection between SFO and European cities. I've had good experiences connecting at Washington's Dulles (IAD), Chicago (ORD), or Amsterdam. Of course, direct non-stop is always preferable, but flights to London seem to be the only flights of this sort that I can find from SFO.
One feature that I have used only once is a layover of several days at a connecting airport. There is a fee for this, but I found it reasonable to pay $100 for an extra few days in Amsterdam.