What abilities should Fighters to not suck outside of combat as much?
I think two strength that could come into play for a potential fighter:
Leadership (outside of combat role). The fighter as the party leader seems to be a common trope, but at least since 3E, no Fighter really was good at that, 4E basically spun off an entire class around that, the Warlord (which is one of the reasons I enjoyed the class so much.), but this is not so much about the combat stuff and more about the story role.
This could mean skills like Diplomacy, Initimidate and Insight. Maybe abilities related to inspiring people to work harder in non-combat situations.
Craftsmanship & Engineering: Not just limited to maintaining or crafting weapons and armor, but to a general eduction on how to craft and build things or how buildings are constructed and designed. This could relate to something like Dungeoneering and Crafting.
Education: Sure, the Wizard is the smart-ass, but a Fighter might also have a strong background of nobility or otherwise entered an officer career. This might cover things like history, nobility and royality, and law. Some overlap with leadership perhaps specifically ot knowledge about warfare and historical battles?
Watchmen/Guard: This could relate very specifically to actual guard duty during campaign, but could also extent into something like Investigation.
Athlete: I mention it, but I think every Fighter needs to have this covered by default to some extent. Their profession just requires high physical excellence.
At the basis, each of these could be simly 1-3 additional proficiencies or maybe even the chance for an expertise, but it might also have something more unique than that, a kind of mini-capstone ability.
If you feel it too much just adding this to the class, it could replace their bonus feats It could be special "Fighter" feats that grant extra benefits?