if thats the case, i really think you just stay put wherever you are staying.. it really isn't worth the trouble as most of the hotels 'nearer' to the airport are less convenient to get to - whereas taxis generally float around in the downtown area a bit more regularly in the early mornings.

As you mentioned you're leaving at 5:45, i can only presume you're flying Delta, which while is a US airline and you'll need to go through a bit more hassle, do the online checkin the night before, get into a taxi at 3:45 and you'll be at the airport by 4. Should be more than enough time.

If you're really really worried, pay the extra S$5.20 the night before and have a cab waiting for you at that time in the morning.. no sweat.

At that time, the traffic is non-existent..