Hey there,
I think it's natural for most applicants to continue to refine their story and messaging as they get more apps under their belts. If you're going at it alone, I'd definitely suggest ramping up the way some of the posters above have mentioned. If you're working with a firm, then this is less of an issue.
What I would say, however, is be careful what you define "dessert" to be. If you're asking if you should apply to a bunch of programs R1 and then apply to your dream school R2, then no. Don't do that. Get your dream program's app in R1. Always. But If you mean within rounds, then definitely start early and start with a couple of apps from other programs. Make sure you manage your time and recommenders appropriately!
If you get to the point of interviews, hopefully you're in a position where you can use earlier sessions as practice. If not and your dream school is scheduled first, then look into prep sessions to make sure you're ready!
Bhavik