Unreal Tournament is a GREAT game for gaming ideas.... I recomment 3 styles.
UT2004 had a game type called "Bombing Run" where you controlled a "bomb" like a football. If you held it, it would regenerate your armor, but the downside is, you couldn't shoot while you had the ball, er, "bomb" you could pass, or LAUNCH the bomb to another player, and the goad was to get it into a portal inside the enemy base. simple game, lots of fun, encouraged teamwork and maneuverability.
Onslaught is a classis... hordes of slow moving bots slowly move toward your Last Stand at the center of the map. keep them away as long as you can, by any means.
Assult is a UT classic, that encourages different objectives to be reached in consecutive order. for the Hawken maps and playing style, it could be more difficult, because UT maps tend to be longer, or more layed out that way, and also, a good mix of open and close quarters. Hawken doesn't cater to this style as much, as we are piloting large mechs, & are not small infantry running around. Perhaps one way to achieve this game type is, we see various destructable objects on the map: Overturned trucks, dumpsters, pieces of debris that can be blown up. Incorporate some special type of this destructable equipment into specific places on the map, and designate them as objectives to be reached. Essentially it's a fast paced version of Domination, whereas there is no question as to the targeted geographic location, and the Attackers must chose to DM, or focus on the objectives. A game type like this could take advantage of the game's multiple, various pathways. Also, perhaps not knowing Objectives A,B, & C right off the bat, but feeding them to the Attackers consecutively, could make it more fun, whereas perhaps letting the Defenders know the objectives could keep them guessing which one the Attackers will focus on first. It's a way to incorporate knowledge of the maps fully, and efficiency of movement is important as fighting skills.