REM cycles typically get longer as the night continues, which is why many find it easier to become lucid in them (more opportunities to notice you're dreaming, as well as rising levels of awareness due to one's natural sleep cycle coming to an end). It's difficult to know how any one person's individual REM cycles work, and the above is only a rough sketch based on the "majority" of dreamers. The only way to know for sure is to have a sleep study done, or use something like a sleep coach, which apparently are unavailable on the market as of now

The reason I choose the third period is so I can leave the rest to DEILDs.
That's actually a pretty good strategy. I've found that Michael Raduga's autosuggestion to induce and notice natural awakenings works really well in the second half of the night. You might check out his book, "The Phase," as it is essentially a guide to DEILD. In short, the instructions are something like:

1. Get out of bed.
2. Urinate.
3. Drink some water.
4. Return to bed.
5. Tell yourself you are going to awaken much more often than normal, and that you will notice these natural awakenings.

He's pretty anal about people doing everything exactly as he says, but the above should be good enough to see some effects. This should be done on a WBTB.