I don't think Ginoza will go after Kougami. He's finally found his place at the PBS, taking care of his team and being a detective, in much the same manner his father was. Also, unlike Tsunemori who is adamant about bringing Kougami in, despite knowing that he'd be locked up for the rest of his life, if not killed outright by Sybil (which is rather intriguing, and hints at an obsession not unlike Kougami's with Makishima), Ginoza seems to be okay with Kougami gone. For him it really was goodbye, and if anything, he wants to help Tsunemori let go too.
When Tsunemori sent Ginoza to find Kougami, she showed Ginoza how much she trusted him - not only with upholding the law, but with looking after her. So Ginoza shouldered the burden of letting Kougami go (thus living with the consequences) and of standing by Tsunemori as she deals with him being gone - again. When he's telling her about Kougami, he's telling her the truth as he sees it. First of all, Kougami got Tsunemori in trouble and endagered her life - again. Secondly, Kougami has changed - he's become harder, more driven and more nonchalant about the ideals he had held first as an inspector and then as an enforcer under Tsunemori's lead.
Thirdly, as Ginoza has finally been able to relax into his true self and open up to others, he has come to care for Tsunemori, love her even. She is precious to him - she treats the enforcers with respect and empathy, she never wavers in her sense of duty, her desire to make society better and safer. She can continue doing the job he couldn't, is stronger than him that way. He would shoulder any burden to help her - even let Kougami get away, thus allowing her to avoid the consequences of bringing him in (having to discipline him, see him locked up or be executed, losing the link she has to Kougami's spirit which helps her maintain her own sanity in troubling times - which I'm sure Ginoza has guessed). When he tells her that Kougami is not worth her attention anymore, Tsunemori picks up on the way he cares for her and wants her to be happy and free.
She also voices what she must have realised long before that final moment in the movie - that Ginoza hasn't changed, meaning that he is still here, not just for the society governed by Sybil and the team of Division 1, but for her - in a way Kougami isn't and hasn't been ever since his obsession with Makishima gained new momentum in season 1. She also lets him know that she has always thought highly of him, even back when he was a stuck-up inspector who gave her a hard time. Back then, he had also sought to keep her safe by telling her to maintain distance from Kougami. And they smile in mutual understanding, caring about and for each other.
I don't think Tsunemori will ever let go of Kougami entirely, since he's helped her grow as a person and a detective, but as she is mindful of her duties and priorities, she will not actively seek him out unless he's connected to an active case.
