So, while I am all for adding some martial maneuvers to fighters, I decided to take a step back. Here is a the Fighter (with some martial maneuvers):
Now, for the exploration/social pillar. Giving some feature at 6th level would help with that, but personally I don't want to give too much since that would take away from other classes more designed than to excel at those pillars. (Again, IMO part of the issue is that other classes are really too good at combat for Fighters to be "the best".)
The bonus ASI/feat normally at level 6 could be used for a feat that would help in other pillars:
Exploration:
- Athlete (easier jumps, better climb speed)
- Dungeon Delver (traps & secret doors)
- Healer (useful if injured during exploration)
- Keen Mind (always know north, recall landmarks)
- Lucky
- Magic Initiate (jump, spider climb, etc.)
- Mobile (+10 speed, Dash ignore difficult terrain)
- Observant (bonus passive scores)
- Resilient (DEX saves since Fighters have STR/CON)
- Ritual Caster (all sorts of spells!)
- Skilled (physical skills, but also Nature, etc.)
Social:
- Actor (infiltration)
- Keen Mind (recall information on NPCs)
- Linguist (languages for interaction, making ciphers)
- Lucky
- Magic Initiate (charm person, etc.)
- Observant (bonus passive scores, would be better if it helped Insight as well)
- Resilient (WIS saves to resist attempts by others to charm or control)
- Ritual Caster (again, all sorts of spells!)
- Skilled (social skills and Perception/Insight)
True, a couple of these involve "magic", but many don't and can certainly help make a fighter more effective in those pillars. But IME we see Fighters mostly concentrate on combat-oriented feats or using the ASIs. This then to me is a player's choice, but they might feel like there is no choice if they want to keep up with combat. So, maybe adding the maneuvers will help allow players to feel free to take these other feats so they can excel in exploration/social.
What about 13th, 14th, and 17th levels?
This is where Fighters can get the features to allow them moments to rival the powers of Wizards. It also fits nicely into the 3rd and 4th tier structure for the people who felt the fixes work better by putting it into the tier structure.
13th Level
This level gives us an extra use of Indomitable. Personally, I think that is pretty weak, and like the idea others have said about making it a "20" instead of a reroll. Maybe at 9th level you get a reroll, but at 13th level it becomes a 20 instead?
However, since this is when casters get 7th level spells, I would rather see a more significant feature here.
14th Level
We have yet another ASI or feat choice. IMO it is a bit late in the game for helping the exploration or social pillars again, but the feats above certainly could be taken for those purposes. Options could also be the second use of Action Surge, or a "Heroic' Surge like I previously posted.
17th Level
Getting another Action Surge use is pretty powerful, but IMO it falls short of what casters are getting with 9th level spells.
However, other classes suffer in power compared to full casters in these tiers, not just Fighters. This is where I also think reducing maximum spell levels to 6th or 7th for up to level 20, and keeping 8th and 9th levels for 21+ and epic play is better and levels the field more between classes.