You do not need to mention a previous rejection simply to prove that you remember it. The cover letter should earn its space by explaining your fit and motivation for the current application.

Mention the earlier process only if it creates a useful story, such as a meaningful improvement since the last application or a relationship with the firm that continued after the rejection.

A sentence like "Since my previous application, I have led X and developed Y" can work when the change is substantial. A long explanation of why the earlier rejection was unfair usually works against you.

Keep the emphasis on the candidate you are now. If the application form asks about prior applications, answer accurately there and let the cover letter focus on present evidence.

I would only bring it up if the new version of the profile is actually meaningfully different.