After beating the main story, you will unlock two new competition routes that can be accessed by speaking to Raimon in the headmaster's office. As with Silvia's routes, there are two rewards per route; one for beating every team, and one for getting an S rank against every team. An S rank is obtained by winning with a 5 goal difference.
During the main game, you can talk to Silvia to request a friendly with teams you defeated in the regional and national Football Frontier Tournament. The final team for each of those routes is unlocked after beating the story mode: Royal Academy for the regional qualifiers, Zeus Academy for the national competition. After beating each of those teams, you're given a rank based on the goal difference you obtained: "S" rank for 5 or more goals, "A" rank for 4 goals and "B" rank for everything else. Beating a team let's you challenge the next one and at the end of each route there's a chest containing two rewards: one for clearing the route, another for clearing the route with an "S" rank on every match. The "S" Rank rewards are manuals for a powerful shot and keeper skill unavailable elsewhere in the game.
To get the Inazuma kit, you have to fill out the player binder. You do not have to scout/recruit every available player, you simply have to encounter them at least once.