Option 3: Jump continues into subsequent turn(s)
The two answers most often referenced seem to conflict, but don't:
"Things like the jump spell don't increase speed. You can jump crazy far, but your speed caps it."
"Every foot jumped costs movement, so you can jump farther than your current speed if you take the Dash action."
- "The trigger you choose for the Ready action must be a "perceivable circumstance" (PH, 193). A caster doesn't perceive turns ending".
Answer 1 doesn't directly answer the question. It addresses "speed of movement", but not "total distance of jump", or the transition between turns. Speed is distance moved over a specific time increment. I presume he read the question as, "Can I use all my Movement, then Jump to increase Distance moved in a single turn?" instead of "Can my jump span multiple turns?"
Answer 2 refers to "caster" because the question involved a spell, but the bigger question was triggering a Readied Action (Spell) to go off "at the end of this round". Non-casters can ready actions, so it follows that no character can "perceive turns ending". (This answer was posted nearly three years later, so if the two do conflict, the most recent rulings hold.)
They don't conflict if you consider that "Speed", "Distance" and "Movement" are not identical, and have specific in-game meanings. Distance: how far a character moves. Speed: distance moved per turn. Movement: maximum possible distance moved in a single turn.
1) Creatures needn't be on the ground when a turn/round ends
"A strong wind ... makes flying by nonmagical means nearly impossible. A flying creature in a strong wind must land at the end of its turn or fall" (DMG p.110)
A creature flying doesn't can end its "turn" in the air, because its movement flows uninterrupted from one turn to the next. Because to characters in the game, turns/rounds don't exist at all.
2) Creatures needn't hover to remain aloft (@JeremyECrawford, July 2015)
"A flyer that lacks the hover trait can stay aloft without moving each round."
A flying creature does not need the ability to stay in one place to stay aloft during the transition from one round to the next. It can remain flying as long as it is in control of itself. This indicates that to the flying creature, it's not moving for six seconds then "waiting" then moving again, it's just moving.
3) Flying creatures remain so unless acted upon
"If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic" (PHB, p.191)
A creature in flight doesn't fall unless one of the listed conditions are met. This reinforces the previous points.
Everything above is consistent with a jump spanning multiple turns
A character that doesn't know its turn has ended has no reason to curtail its movement to something that can be finished in a six-second span. It can jump further than its Speed, but can only move up to its Movement limit in a single turn.
If this were otherwise, no one in the in-game world could ever throw a ball, shoot an arrow, toss a javelin, etc., if the distance from start to finish took longer than six seconds at its speed of travel: the projectile would simply stop mid-air and drop to the ground. Everyone in the world would "perceive turns ending", since anything without flying or under the control of a spell would stop moving and drop to the ground in a predictable steady six-second pulse.
Another angle: Turns don't exist outside of combat. They don't exist to characters at all, only players' minds. If anyone or anything in their world could ever make a long jump or long throw (archers firing arrows at high angle to increase range, giants throwing stones long distances), but couldn't do so while in combat, they would be aware of the six-second "turn/round" pulse, and that doesn't add up.