You can still have interesting stuff with more than two teams!

Players compete for points, and get 50 points if their team wins, but get points for other things: negative points for being killed, positive points for sub-objectives that don't particularly help their team (picking up items hidden throughout the playing field, standing on tall things...).

For extra fun, give each player a random secret objective (including a "you actually play for the other team, and get scored accordingly" card).

Or for a totally different feel, have an "accomplish objectives" game (find hidden items, or bring back water to your jar with a spoon, the one with the most wins), but also give them paintball/airsoft guns, with rules on death/respawning etc.