There is a Tsimoridis and a Tsimaratos listed at ellisisland.org for the early 1900s. Try there. I tried the census for Tzimoratos and the two versions above and found nothing.

You could also find the surname navigator at cyndi'slist.com, under greece, and type in Tzimoratos, which will give you a page that is the Greek Telephone directory, listing all the Tzimoratos names in the book.

Good luck!