Most states accept the S corporation election (2553) without further issue.
With respect to New York, you most likely will not have to register separately since you have done so with respect to your LLC articles of organization and the state should accept your federal election. However, you might want to confirm this with the state.
See https://www.upcounsel.com/new-york-s-corporation-election
See also https://www.nolo.com/legal-encyclopedia/annual-report-tax-filing-requirements-new-york-llcs.html
