A new COI is required
After changing/updating the particulars on the IC, apply for a new Certificate of Identity. Updating the IC does not automatically rewrite the travel document.
Preparation
Use the separate COI application process with the updated IC, relevant photographs/forms and most recent COI. For example, a changed name should be consistent across the updated identity record and the new travel document.
Official ICA source
Written independently from ICA’s English original. The linked source supplies the complete document and subsequent updates.