How does moving silently with Greater Invisibility after taking an action work?

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.