I think one area that's ripe for a fighter is tool use (and yes I know everyone can gain tool usage, but the incentive for the Fighter to gain them so they have more to do when out of combat is higher than for other classes). It's something which can be learned in downtime, and which has potentially much wider application than I suspect a lot of games use/exploit to their most advantage. Xanathar's Guide went a long way to helping out with this.
For example at level 3 a Battle Master gets "Student of War", where you gain proficiency with one type of artisan’s tools of your choice. I picked Carpentry Tools, and this is on my Character Sheet with all my other abilities:
Carpentry: Enables a character to construct wooden structures; house, a shack, a wooden cabinet, or similar items. Components. saw, hammer, nails, hatchet, square, ruler, adze, plane, chisel. Xanathar's Pg 78
History. Aids you in identifying the use and the origin of wooden buildings and other large wooden objects.
Investigation. Additional insight when inspecting areas within wooden structures, because you know tricks of construction that can conceal areas from discovery.
Perception. You can spot irregularities in wooden walls or floors, making it easier to find trap doors and secret passages.
Stealth. You can quickly assess the weak spots in a wooden floor, making it easier to avoid the places that creek and groan when they’re stepped on.
Fortify. With 1 minute of work and raw materials, you can make a door or window harder to force open. Increase the DC needed to open it by 5.
Temporary Shelter. As part of a long rest, you can construct a lean-to or a similar shelter to keep your group dry and in the shade for the duration of the rest. Because it was fashioned quickly from whatever wood was available, the shelter collapses 1d3 days after being assembled.
And then the list of four DCs listed in Xanathar's.
What this means is I now essentially have proficiency in History, Investigation, Perception and Stealth when those checks are made concerning certain activities over or with wooden structures, objects, floors, walls, etc.. And I can help block a door during short and long rests, and help out with shelters. All good stuff for out of combat activities.