"Convincingly" is an adverb, and adverbs describe how something is done. They don't typically have comparative forms.
To express a greater degree of conviction, you could use phrases like:
* More convincingly
* Highly convincingly
* Even more convincingly
* Most convincingly
For example:
* "He argued his case convincingly."
* "She presented her proposal more convincingly than the others."
* "The evidence was highly convincing."