As you noted in your answer, the Cavalier, a fighter subclass from Xanathar's Guide to Everything (p. 30-31), has this feature:
Starting at 18th level, you respond to danger with extraordinary vigilance. In combat, you get a special reaction that you can take once on every creature’s turn, except your turn. You can use this special reaction only to make an opportunity attack, and you can’t use it on the same turn that you take your normal reaction.
This lets you make one extra opportunity attack on every creature's turn (except a turn in which you use your regular reaction).
Alternately, Unearthed Arcana: Light, Dark, Underdark! features an additional Fighting Style option (for fighters, paladins, and rangers who have the Fighting Style class feature), the Tunnel Fighter fighting style:
You excel at defending narrow passages, doorways, and other tight spaces. As a bonus action, you can enter a defensive stance that lasts until the start of your next turn. While in your defensive stance, you can make opportunity attacks without using your reaction, and you can use your reaction to make a melee attack against a creature that moves more than 5 feet while within your reach.
This fighting style allows the character to make as many opportunity attacks as they can without using their reaction, and additionally allows them to use their reaction to attack a creature that moves around them within their reach - all for the cost of just a bonus action.
However, keep in mind that UA is playtest content, and may not be balanced. The two Fighting Style options from this 2015 UA (Close Quarters Shooter, and Tunnel Fighter) have not been published in any 5th edition book so far. It is possible that Wizards of the Coast deemed them unbalanced in their current form, or that Wizards decided the fighting styles required further tweaking/iterations before publication.
Especially by comparison to the Cavalier's 18th-level feature, the Tunnel Fighter fighting style seems quite overpowered, especially for a 1st-level fighter feature (or a 2nd-level feature for rangers and paladins).