With[{x = a}, Hold[With[{a = b}, a + 2 x]]]
(*==>Hold[With[{a$ = b}, a$ + 2 a]]*)

With[{x = a$}, Hold[With[{a = b}, a + 2 x]]]
(*==>Hold[With[{a$ = b}, a$ + 2 a$]]*)

With[{x = a}, Hold[With[{aa$ = b}, aa$ + 2 x]]]
(*==>Hold[With[{aa$ = b}, aa$ + 2 a]]*)

With[{x = a}, Hold[With[{aa$$ = b}, aa$$ + 2 x]]]
(*==>Hold[With[{aa$$$ = b}, aa$$$ + 2 a]]*)

With[{x=var1$},...blackBoxCode...]

With[{x=var2$},...blackBoxCode...].