-- UPDATE --
In reflection, I now understand that the Jump action doesn't remove jump from movement, but instead behaves more like the dash action in that it is something a character can do on their turn as well as move. It's complementary. This changes things a bit, so I have updated my post below to follow suit.
In the recent Unearthed Arcana for One D&D, designers have introduced Jump as an action.
This new action supplements the action economy formalising what many game masters would allow at their table - that being if someone wanted to move and then jump, they can naturally use their full movement and then take the jump action for extra distance.
This really highlights an underwhelming development with the 5e rule set, in that the designers are continually grasping at water; meaning each "design problem" of the rules is being solved by providing more rules of increasing detail.
Do we need a dash action? Do we need a jump action? Could these both be better handled by a general rule that allows for a character to take a skill action. Don't Dash and Jump align to the athletics and acrobatics skills already (player's choice).
Interestingly OneD&D also codifies the search action, and aligns it to the skills of perception, investigation, even history, etc. based on the subject (or topic) of the search. Again, does this need to be it's own action in combat, or is this just a use of a skill. I think the later.
If we adopt a skill action approach we can simply get rid of Dash, Jump, and Search, and even Hide as it is a stealth skill action anyways. This to me makes more sense as it provide a general rule to fit all skills in the game within the action economy. There is already a catch all action in combat called the "Improvising an Action" action (a bit wordy!). If this was renamed to "Use a skill" this would mean improvising would be for the outlier circumstances.
This would reduce combat actions to: Attack, Disengage, Dodge, Help, Ready, Use Magic, Use a Skill, Use an Object, and Improvising an Action. This reduction is better than bloat for me.
Now with the general case solved for, I think the martial character types need some love when it comes to jumping, and moving in general when they take the attack action.
Here is a suggestion:
"When you take the attack action on your turn with a natural or proficient weapon, you can also include an athletics or acrobatics movement either before or after one of your attacks. You need to be proficient with the skill used, and if the game master determines you need a skill test to succeed they will provide you with a DC to fit the circumstance."
Breaking this down: it means a martial character can do a jump and attack, swing from the chandelier (climbing action) and attack, a tumbling attack, etc. It's something simple but adds tonnes of fictional fun for the fight.
If a skill test is needed, then this is easily included by the game master. A failure could easily foil the possibility of the attack, or leave the character vulnerable. So I would suggest generally easy DC's - e.g. 10 or 12; as this would be something that a martial character should have trained. I guess this will depend on the level of fantasy at which the game is played - 5e usually has this dial set to high fantasy with the character's being heroes.
Interestingly, there is no easy place to insert this fix in any of the classes.
A lack-luster resolution would be to add it to rage for Barbarians, Martial Arts for Monks, Favored Foe for Rangers, perhaps even to Cunning Action for Rogues. But it just doesn't fit anywhere in the fighter's features.
Martial Prowess. If you are proficient with martial weapons or martial arts you gain the martial prowess feature.
When you take the attack action on your turn with a natural or proficient weapon, you can also include an athletics or acrobatics movement either before or after one of your attacks. You need to be proficient with the skill used, and if the game master determines you need a skill test to succeed they will provide you with a DC to fit the circumstance.
For Rogues, they should get "Use a Skill" as a cunning action. This might overlap with a Thief's Fast Hands feature - but that needs a fix anyways.
And this is done. Enjoy.