This question could be simplified with "How does moving silently away from enemy on same turn after casting Invisibility work", but I'll present the full scenario in case it makes a difference:
A PC, who does not have special bonus actions, is under Greater Invisibility spell (or other equal effect) which allows actions without breaking invisiblity. At the start of their turn in combat, they are 10 ft (one empty 5 feet square between them on the grid) away from an enemy. They then move 5ft, uses their entire attack action to do a melee attack on the enemy (who does not go down). Then they tell the DM "I want to move back 5 feet, then use rest of my movement silently move away, so the enemy doesn't know where I actually go."
What should DM respond, by the rules?
The PC can't use hide action, or any action for that matter, because they already used their action to attack. Movement rules don't have anything special about silent movement as far as I could find. But it also seems quite unreasonable and suspension-of-disbelief breaking to say "sorry, but you are utterly unable to move silently at this moment".
The reason the PC wants their location to not be known is to prevent the enemy from approaching to melee range and hitting them, in this case. The reason could also be the enemy using a spell like Moonbeam, or PC wanting the enemy to waste a blind ranged attack at PC's (who could be in full cover now, even) last known location.
It's of course easy to make a custom ruling here, involving some combination of PC Stealth, enemy Perception, and reduced movement rate, but I'm interested in what the rules say, including any language which supports handling this as part of a custom/improvised action, even though the PC doesn't have an action to spare.
Best Answer
In 5e, your 6 second combat round isn't "do all of A then do B".
When you move and attack, you are doing it all at once; you are attacking as you approach, ducking and weaving, and attacking as you retreat.
30 feet isn't very far; a human can easily cross that distance in 3 seconds.
What they did was move at a full run 5 feet forward, duck and weave and attack for 3 seconds, deke out 5 feet (again at a full run) while still attacking. At best they have 2 seconds of time to go from a full sprint to some kind of silent movement, well enough that an enemy doesn't have an idea where they are?
Extremely difficult, even when invisible.
Greater Invisibility offers automatic advantage on all attacks and disadvantage on all attacks by an enemy. This covers "I can only hear you, I can't see you" advantage already. If the player wants to upgrade this to "you have no clue where to shoot", they need to spend the action resource to take the hide action.
Moving quietly, breathing quietly, and moving unpredictably are covered by the cost of using an action (for most PCs). Your speed is limited to 30' per round, as half of your focus is on not being noticed.
For those with special training (like a Rogue), they can do this while also attacking. They can step in, invisible, stab someone, bonus action hide, then move away to an unknown location. Enemies now have to both guess where the Rogue is, and even if they do still have disadvantage on attacks.
Combatants are already expected to be putting their best effort into dodging, avoiding being spotted, etc, given that they are otherwise occupied by their main action. "I move at half speed to be more defensive", "I move at half speed to move quieter" is covered in 5e rules by expending your action to do that.
Now, maybe you feel that PCs aren't competent enough as is.
If you feel that PCs don't have enough capability to do things on their turn, hand out a 2nd action to every PC. Then that PC can deke in and attack, then spend an action to hide as they retreat. That will also require some work to fix up the game balance, but at least it won't make a powerful spell (Greater Invisibility) even stronger.
Invisibility in 5e is limited by design and by experience. It makes fulfilling the requirements of hiding easy, but it intentionally does not remove the action cost to do it.
The dual advantages and disadvantages of invisibility makes it a strong spell as it stands.
If this bothers you, and you think being completely invisible should be better than 5e makes it, then rather than making invisibility effects 2x or more stronger, make add narrative shimmers to invisibility effects; when you move, while invisible, difficult-to-spot shimmers occur. PCs who don't spend an action hiding can be tracked by such shimmers, and even ones that are hiding have to win a stealth vs passive perception check to not be spotted.