Just because you hit SP doesn't mean you can get to LD. The reason it is so hard without any sleep is because your body skips REM sleep when it enters sleep on the first cycle. LD's happen in REM. It's true that actual SP happens in REM, but the body hits a SP like state when it enters sleep initially (skipping REM).

If you are really tired or well trained you can hit REM sleep without going through a full 90 minute cycle, but the general rule is that you don't have REM until you get back up to it at the end of your first cycle.

After 6 hours you spend much of your time in REM sleep. That is why it is easier after 6 hours.