Problem being, every other Starship-class commander is shown to be older or higher ranked than Kirk.
Plus, there isn't a whole lot of time for Kirk to have commanded another ship. If Starfleet is a five-year school, then you have a man who graduated at 23. We know from The Deadly Years
that Kirk is 34 and likely took command of the Enterprise at no later than age 31. Which leaves just eight years for him to climb the ranks and I can't believe he was accelerated rank-to-rank every step of the way.
Unless someone has something more than Dehner's line from the actual episodes, then I tend to believe that the Enterprise was Kirk's first command.