No, you do not need to re-enroll. I also used my green card to enroll. After I Naturalized and eventually received my US passport, I contacted TSA Enrollment by IDEMIA (Because that is how I originally enrolled) at 855-347-8371. If you enrolled via CLEAR you can call 888-831-9070, or if you enrolled via TELOS the number is 833-777-1811.
After I spoke with a representative, they had me email TSAEnrollmentDocuments@tsa.dhs.gov with my US passport and driver's license along with my information:
First Name:
Middle Name:
Last Name:
KTN:
External ID # (which they gave me during the call)
Then they confirmed my information was changed within 72 hours.
Hope that helps.