they should *look* the day before, but your flight will actually be on the day you intend to travel. (that is, if you want the flight on Friday at 00:50 ask them to check flights on Thursday. in doing that, they'll see the friday flight. or so the theory goes)
easy way to check is to confirm the arrival time (although US often gets those confused!).
Thanks- I asked them to look the day before and they said 'the Thursday flight is available'. I asked whether that lands thurs am or Fri am (as this would solve the question) and they said 'it doesn't say but I assume Thursday because it would land same day'