I know that attacking while invisible reveals my location. But what if I move after the attack? What if I fly (using winged boots)? Will my (new) location still be known then?
When you try to hide, make a Dexterity (Stealth) check. Until you are discovered or you stop hiding, that check's total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence. [..] An invisible creature can't be seen, s0 it can always try to hide. Signs of its passage might still be noticed, however, and it still has to stay quiet. (PHB 177, emphasis mine)
After you are noticed your previous stealth check is no longer "valid" and you need to make another by taking the Hide action. Other signs of your passing give you away, eg. footsteps. Being invisible only means you do not need to find cover or obscurement to do it.
Jeremy Crawford, a D&D 5e game designer, explicitly says that invisibility doesn't hide you:
The invisibility spell doesn't automatically hide you; you still make noise.
However, you can always try to hide while invisible (except for some rare cases), as the Szega's answer suggests:
An invisible creature can't be seen, so it can always try to hide.
This requires an action though.