You can use Misty Escape if you take damage.
Misty Escape states:
When you take damage, you can use your reaction to turn invisible and teleport up to 60 feet to an unoccupied space you can see.
You can take damage out of combat, so there is no reason you cannot use Misty Escape out of combat.
“Until your next turn” is six seconds, because a round is six seconds long.
The rules for combat state:
A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.
So when Misty Escape says:
You remain invisible until the start of your next turn or until you attack or cast a spell.
It means that you stay invisible for the next six seconds.
You are able to take reactions outside of combat.
Just in case there is any question about taking a reaction outside of combat, I’ve copied my argument from this answer
As evidence of being able to take a reaction in response to a relevant trigger outside of combat, consider the spell feather fall:
Choose up to five falling creatures within range. A falling creature's rate of descent slows to 60 feet per round until the spell ends. If the creature lands before the spell ends, it takes no falling damage and can land on its feet, and the spell ends for that creature.
Feather fall has a casting time of 1 Reaction,
which you take when you or a creature within 60 feet of you falls.
I think it quite obvious that the act of falling does not initiate combat or require rolling initiative, and so there is a clear sense in which feather fall is intended to be usable outside of combat. Therefore, you have your reaction available outside of combat.