I have been looking for a car too. My budget was this much but now I am at 8.5 and still searching.
In 6.5, you can get City exi/exis. BUT its very very difficult to find a clean ride. If you do, its an awesome car.
Cultus 2006-07 is also an option. You can get some in well maintained condition.
Alto is another option.
Santro 2006-2008 all fit in 5.5k budget.