The rules for rolling team base dial is as follows

At the end of your turn, you may roll a d6 to get abilities that will be more useful. If the Team combat icon ability is showing at the end of your turn you must make this d6 roll.

So let's say I choose to roll a d6 at the end of my turn, and I end up on a slot with the Team combat icon showing. Since it's still the end of my turn, does the 2nd part kick in, forcing me to roll another d6 to click it again?

If so, then when can I use the team combat icon?

You only ever roll once at the end of your turn, regardless of what is showing after you roll. It's just that it's optional to roll if you don't have the symbol showing and mandatory if you do. So, if you roll on to one of those clicks at the end of your turn, you'll have the symbol for your next turn, but you must roll again at the end of that turn.

There is a line in the Player's Guide that says you can only ever do something once when it would apply. If you already rolled, you would not roll again that turn.