RAI not automatically
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, The creature lands prone, unless it avoids Taking Damage from the fall.
It is certainly true that this passage, taken literally, means that someone who avoids taking fall damage does not land prone. It should be clear that the intent is, however, that someone who starts the fall not prone does not then become prone as a result of the fall, because they did not take damage. This fits the context of a typical fall and makes sense; usually someone is not prone when they begin their fall and they are knocked prone as a result of hitting the ground.
It would be strange indeed for someone to start a fall prone and then have the fact that they did not take damage somehow force them upright at the end of the fall. Clearly that is not the intent of the rule, which assumes that they are not prone initially. The mistaken assumption about initial conditions resulting in an illogical conclusion is perfectly paralleled by an old joke in English:
A patient asks a doctor, “Will I be able to play the piano after the operation?”
“Certainly,” the doctor replies.
“That’s great! I wasn’t able to play the piano before,” the patient remarks.
For example, suppose a creature is wearing a ring of feather falling when they are Paralyzed. Then, they are knocked prone and pitched off a high balcony. A second creature is Petrified, toppled, and rolled off a nine-and-a-half-foot drop. Yes, a literal interpretation of the rule means that somehow they both will remain prone for their entire fall but will then both miraculously land upright when they hit the ground. However, most DMs would rule this were nonsensical applications of the rule.
Even the appeal to Szega's high-scoring answer about using teleportation to counter the prone condition is, in my opinion, misplaced. Their argument is not that teleportation allows one to re-orient in space (which at least is plausible within the narrative) but rather that a literalist interpretation of the rule above says that someone who starts prone but doesn't take damage in a fall will be placed upright - not by the magic of teleportation, but by the power of the rule. Yes, this is literally true RAW, but it is narratively silly.
The rule should read:
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell. The creature is knocked prone upon landing, unless it avoids Taking Damage from the fall.
Applying this common-sense revision would make the rule align with what most DMs are going to rule anyway.
Electing to use movement
If simply falling without taking damage is not enough to remove the prone condition automatically, there are multiple ways within the rules to do so; however, they would either require you to "stand up from prone" or otherwise use extra movement.
From your starting position prone atop the table, you could simply stand up on the table (costs Move of half your Speed) and then leap down, landing on your feet since the fall was less than ten feet and you didn't take damage.
If for some reason you didn't want to stand up while on top of the table (say, you were under ranged attack), you would need to move off the table before standing up. We know that moving while prone has to be done at a crawl, and that crawling costs an extra foot for every foot moved. So you will crawl to the edge of the table, move off, fall while prone, not take damage since it is less than 10 feet, land prone (supposing, as I contend, the mere act of falling doesn't force you upright), and then stand up (at the normal cost to stand up).
Finally, if you don't want to fall, you could crawl to the edge of the table and lower yourself down. I think this could be considered 'climbing', since you would be pushing off of a vertical surface. Interestingly, while the PHB tells us that climbing is a "Special Type of Movement", it doesn't actually tell us what climbing is or when it should be used (or how it interacts with prone), so that will be a decision for your DM. If it were me, I would be comfortable saying that paying the movement penalty for climbing down your curiously-high table would result in you being vertical, and thus not prone, by the time you reached the floor, without having to have separately paid to stand up. That would be a ruling, though, not a rule.
Any of these three methods would get you off the table and on your feet with no chance of failure but with the expenditure of a considerable amount of your move. But I suspect you are after something flashier.
Petit Parkour?
Your question (and subsequent answer and comments) seem to indicate that what you would really like to do is use the act of falling to re-orient yourself so that you start prone but land on your feet, without having to have expended any extra movement to "stand up from prone". This is entirely reasonable if you can get your feet to 'fall first', before the rest of your body. But it should take a certain amount of skill to do well, that is, without hesitation or losing any movement. In this case, what we have is a 'skill challenge' - you are suggesting something that some characters should be able to do better than others, and that won't always work.
Consider the definition of a Dexterity (Acrobatics) check:
Your Dexterity (Acrobatics) check covers your attempt to stay on your feet in a tricky situation, such as when you're trying to run across a sheet of ice, balance on a tightrope, or stay upright on a rocking ship's deck. The DM might also call for a Dexterity (Acrobatics) check to see if you can perform acrobatic stunts, including dives, rolls, somersaults, and flips.
The situation you describe, using a controlled fall off a high table to become upright from prone without having to spend movement to do so, is for me a textbook case for a low-DC Acrobatics check, with success operating just as you would like it to. If you were in Heavy armor or encumbered while doing this, I would either set the DC higher or call for a Strength (Athletics) check, treating it as an attempt to "pull off a stunt midjump."
My guiding principle here as a DM is that what you are trying to do is reasonable and possible, but it is not an automatic consequence of the rules of movement or falling. Thus you don't get it for 'free'; either you spend movement to make sure it works like you want, or you rolls the dice and you takes your chance.
