mowing 6" high

Because of construction and rocks I need to mow 6" high rather than 4". I'm looking for a cheap hand mower that can mow at 6" level.
Please help.

Sounds like string trimmer work. Or one of those wheeled DR trimmers.

Or add taller tires to your push mow.

We call them slasher down here I think you lot call them rough cut mowers.
At 6" you have to be really careful about where you put your tootsies, if you want to keep on dancing.
The local shire run some tall walk behinds with a chain curtain at the back and rubber flaps on the non discharge side.
So if you go for bigger wheels condiser some side protection.
Also note when the deck is that high clippings will come out all around the side not just the chute so it is long thick trousers and steel capped boots.