You could get 5 additional rounds.
tl;dr The unit of concentration is a round. Breaking concentration in the final round gets an uncontrolled demon for up to 6 rounds including the one in which concentration was broken.
Breaking concentration immediately results in an uncontrolled demon.
The spell description details what happens when the demon is no longer controlled:
On a successful save, your control of the demon ends for the rest of the duration, and the demon spends its turns pursuing and attacking the nearest non-demons to the best of its ability.
The character concentrating chooses to break concentration on their turn during the last round of the spell's normal duration. If the demon's initiative is after the character, its is going to attack. If the demon's initiative was prior, it's already used it's turn. Either way the current round it is now uncontrolled.
The special case of ceasing concentration does add additional rounds:
If you stop concentrating on the spell before it reaches its full duration, an uncontrolled demon doesn't disappear for 1d6 rounds if it still has hit points.
Rolling a 6 when you cease concentration means that it does not disappear for six rounds. Since the round you cease concentration is the first round it is uncontrolled, this could net 5 additional rounds it remains beyond the duration of the spell.