From the official synopsis we know a "new government organization will go toe-to-toe with S.H.I.E.L.D.". The synopsis later mentions the ATCU or Advanced Threat Containment Unit, which I assume is the government organization referred to. I’m hoping this group will be attempting to track down and lock up any new Inhumans throughout the season. They will essentially be the personification of Simmons hardline approach to powered people that she adopted in Season 2. In order to lock up the newly powered people they will build the MCU version of the Project 42 prison. Meanwhile Skye and Coulson will be recruiting for the Secret Warriors, eventually forming a team. This will be separate from it’s parent organization S.H.I.E.L.D and the government. At the same time Ward and Hydra will be looking to recruit Inhumans. This three way race will result in lots of clashes over the course of the season.
The synopsis for Civil War states that “after the events of Avengers: Age of Ultron, another international incident involving Captain America and the Avengers results in collateral damage, prompting politicians to form a system of accountability and a governing body to determine when to call in the Avengers, which results in the fracturing of the team while they attempt to protect the world from a new enemy”.
I think the international incident will follow on from the finale of AoS3 and be the final straw for the public and the government. The ATCU, its leader(s) and Project 42 should all appear in Civil War and it is with this organization that Stark and his supporters align themselves. Thunderbolt Ross could even use the Inhumans that had been locked up in Project 42 over the course of AoS3 to form the Thunderbolts and help track down any rogue heroes.
Meanwhile the ideals of the Secret Warriors will be shared by Captain America and his supporters, so there is chance for a crossover there – even if it is something minor such as using a Secret Warrior base as a hideout etc. However, if Marvel want to increase the numbers on both sides and supply cannon fodder for an epic final fight then using the Inhumans introduced in AoS3 would be the best way to do so - Cap’s team and the Secret Warriors Vs Iron Man’s team and the Thunderbolts!