It seems you can use a jump as part of a move action and attack as a separate standard action in 3.0/3.5
First, let's look at the 3.5 jump skill
Jump (relevant parts)
If you have ranks in Jump and you succeed on a Jump check, you land on your feet (when appropriate). If you attempt a Jump check untrained, you land prone unless you beat the DC by 5 or more.
The part about, "you land on your feet", is simply describing that you don't land prone. No house rules required.
High Jump: A high jump is a vertical leap made to reach a ledge high above or to grasp something overhead (The last part is descriptive text and it's not exhaustive, notice the following "If's".). The DC is equal to 4 times the distance to be cleared. If you jumped up to grab something, a successful check indicates that you reached the desired height. If you wish to pull yourself up, you can do so with a move action and a DC 15 Climb check. If you fail the Jump check, you do not reach the height, and you land on your feet in the same spot from which you jumped.
The move action+DC 15 Climb check that, "pulls yourself up", is separate from the move action made to jump. Implying that you can jump as part of a move action and use another action midair. One might say, "Jumps don’t “end” in the air". There is nothing in the 3.5 rules that implies you can't end a move action in the air. As the 3.5 Jump skill implies.
Here's how this would look in a turn. My 3rd level Monk with 40 ft. movement is going to attack an imp flying 10 feet in the air. The monk is 20 feet away from the imp.
As a move action, I get a "running start, which requires that you move at least 20 feet in a straight line before attempting the jump. (This is different from a full-round Run action.)". I make a DC 20 jump check (as part of this movement). I succeed, so I've reached my desired height. I now threaten the imp. I could try and grapple him as a standard action, but I'd rather use my signature "Kim Dragon Kick", ;-) instead, as a standard action. I hit and kill the imp, falling 5 feet to the ground. Even if the Imp lived I would not have provoked an attack of opportunity, due to it being tiny.
If you'd like to say that the Monk would start falling first after the move action, that's fine, as you could still make the standard action attack according to the Jumping down rules, with Battle Jump clarifying (see below).
Let's take a moment to look at the controversial Long Jump section
Long Jump: A long jump is a horizontal jump, made across a gap like a chasm or stream. At the midpoint of the jump, you attain a vertical height equal to one-quarter of the horizontal distance. The DC for the jump is equal to the distance jumped (in feet).
If your check succeeds, you land on your feet at the far end. If you fail the check by less than 5, you don’t clear the distance, but you can make a DC 15 Reflex save to grab the far edge of the gap. You end your movement grasping the far edge. If that leaves you dangling over a chasm or gap, getting up requires a move action and a DC 15 Climb check.
I believe the line "If your check succeeds, you land on your feet at the far end," is not meant to mean you can't take other actions after making a long jump as part of a move action. It's, if you just make a long jump and succeed, you will land on your feet (i.e. not prone) at the end of it. This way, not only does it work with the High Jump rules. It supports what is considered the "normal" rules by the Battle Jump feat and Acrobatic Attack special ability, as seen below.
Acrobatic Attack (Ex): At 5th level, if the blade dancer attacks by jumping at least 5 feet toward his opponent, jumping down at least 5 feet onto his opponent or swinging on a rope or similar object into his opponent, he gains a +2 bonus on attack and damage rolls. The blade dancer must make a jump check; if the result is less than 5 feet, he cannot use this ability on that attack. If the result is greater than the distance between the blade dancer and the opponent, the blade dancer can limit the distance to that between himself and the opponent as a free action.
The parts in bold directly above appear to be normal, except the "+2 bonus on attack and damage rolls". Meaning you can jump at your opponent and attack them.
This needn't lead into, "Mario Bros.-style double jumps", as there is a big difference between, jumping & attacking or pulling oneself up, and jumping again while already in the air!
Jumping Down: If you intentionally jump from a height, you take less damage than you would if you just fell. The DC to jump down from a height is 15. You do not have to get a running start to jump down, so the DC is not doubled if you do not get a running start.
If you succeed on the check, you take falling damage as if you had dropped 10 fewer feet than you actually did.
Benefit: You can execute a charge by simply dropping from a height of at least 5 feet above your opponent. For example, a ledge 10 feet above the floor of a cavern would suffice for jumping on a Medium-sized creature, while a ledge 15 feet high is required for a jumping on a Large creature. You can't jump from more than 30 feet above your opponent, nor can you effectively battle jump while under the influence of a fly or levitate spell or effect, as you have to hurl yourself down on your foe.
If you hit, you can choose either to deal double damage with a melee weapon or natural attack or to attempt a trip attack. You are treated as one size category larger than normal if you try to trip your opponent with the battle jump. After you attack, you take falling damage as normal for the distance you jumped. You are entitled to a Jump check (DC 15) to take less damage, as if you had fallen 10 feet less than you actually did. If you fail this Jump check, you fall prone 5 feet from your opponent.
You can also use Battle Jump to begin a grapple attempt instead of making a normal attack. If you do, you are treated as one size category larger than normal for the first grapple check following the battle jump.
Normal: Anybody can try to jump down on an enemy, but it is not considered a charge, and they do not gain double damage or the size bonus for the ensuing attack. (Unapproachable East, p. 42)
This clearly implies that normally you can use a move action to move and jump down as part of that movement, then take a standard action to attack your foe.
Making a long jump and grabbing a rope hanging above you in the middle of a chasm, or long jumping and then grappling a flying creature is something the rules should and do support, without the need for...a special ability, three feats, or employing magic!
Using Pathfinder rules as dominant across the board (which one should never do):
Pathfinder disallows this...
Leap and Bound (Ex) (Chronicle of Legends pg. 7 Release date: May 29, 2019): A vigilante with this talent adds his Strength bonus on Acrobatics checks in addition to his Dexterity modifier. He is always treated as having a running start when jumping, and his high jumps are treated as long jumps when determining the DC. When the vigilante jumps, he does not fall until the end of his turn, allowing him to attack or perform other actions in midair. If the vigilante grapples a creature capable of bearing his weight, he does not fall, instead remaining adjacent to the creature as it moves. A vigilante must be at least 10th level to select this talent.
This clearly implies that without this talent, or by another means, you cannot perform other actions midair after jumping! So long to...pulling yourself up after a high jump, grabbing a rope, etc.
If your GM takes this approach the following might help you...
Better yet, talk you GM to adopting this interpretation of the 3.0/3.5 rules for Pathfinder! Monks... nay, all melee combatants will thank you everywhere!