I enter (or not) from the New Sweeps listing, and have never even looked at the expiring sweep list since I joined SA.
If you enter a single entry sweep there is no advantage to entering at the beginning or the end - the odds are the same. You're at a distinct disadvantage for those sweeps that allow multiple entries (daily, instant, weekly etc.), as you will only have one entry when you could have had many more had you started entering from the New Sweeps listing. A daily entry sweep for 45 days would give you 45 entries rather than just the one you'd get from the expiring list. You should check the New Sweeps listing again the next day, so that you catch any late additions.