The relation between Accelerator and Mikoto starts with a fight, in the third novel.
He was the main plan of Aleister, and the core of the project to create the first Level 6, killing 10.000 Misaka imouto.
After the death of a lot of them, Mikoto decided to faces him, and she was ready to die in order to save all the other sisters. But thanks to Touma, Accelerator is defeated and the imouto are safe, and after that moment, they don't have the chance to meet again.
It's in Hawaii, in NT3, that they finally share moments together. It's the first time since months, and Mikoto doesn't know about Accelerator's actions to protect her sister.
That was the time she met Misaka WORST for the first time, and she doesn't know about Last Order, so they have a lot of things to talk about.
In fact, they fights together against GREMLIN, and while they kick their opponents, they decided to have a meeting soon, to talk about all the things that they don't kwnow of each other. Mikoto needs to know about Last Order, and there are a lot of things that she wants to know from him (especially WHY he's protecting his imouto and he lives with them, I think), also he needs to speak with Mikoto about all the problems they have... as a family.