Can we do "anniversary-based" 360 reviews? For instance, we want to do 90-days review for new hires, or review each person exactly one year after he is hired.
We address this question in our article, here.
In addition to the above answer, it may be worth noting that you may not require a full-fledged 360 cycle just to give feedback to a few specific individuals.
For instance, maybe it's sufficient to tell the people who should provide feedback for a new hire to simply use the "write feedback" feature on the homepage, and share their insight with that person (and/or their manager).