(What a mess! Ten months, 31 or 30 days each, 10 day weeks, one day of 'weekend', and 51 days left over at the end of the year! yuck!)
After Caesar's assassination, Mark Antony had Caesar's birth month Quintilis renamed July (Iulius) in his honor.
So July wasn't added, just renamed.
And more worserer:
In 8 bc, the plebiscite Lex Pacuvia de Mense Augusto caused Sextilis to be renamed August (Augustus) in his honor.
8 BC, well within the 'fuzz' of the uncertainty of the exact year of His birth.
With further chaos caused by the insertion and removal of intercalary months and near random leap days...