Noooooooo. You do not have to chase a ring and in fact I don't think anybody would recommend it. It usually does have thin/non-existant rings I think anyway. Just scrape the bark off the back and you're golden. I have yet to successfully make my first Hop Hornybeam but it is in the works and I have heard they make very good, fast bows. Some I have seen like it as good as Osage anyway.
CP