Yes! You can run as many catalogs concurrently on your public-facing site as you wish.
The start and end dates you set for each catalog will dictate when the catalog displays on your public-facing site. You can run catalogs with exactly the same dates, or you can stagger them by having one catalog start before the previous catalog is scheduled to end.
Of course, if there are individual classes in a concurrent catalog that you're not ready to display yet, you can turn individual classes "on and off" by marking them as "Unlisted." The catalogs will continue to run concurrently, but any classes that you're not ready for the public to see will remain hidden until you switch their visibility setting back to "public."