I just had a run of 500 boxes made from SBS , automatic bottom, 6.75x3.0x3.0 done for $2.79 each with a $600 die charge.
You want a bigger box made from a black e-flute (I'm assuming e-flute, c-flute would be clunky for a 1 inch box). The black e-flute will likely be a custom run requiring hundreds of pounds of material. You may get lucky that a manufacturer may have some extra, but I wouldn't count on it.
fwiw, I use American Eagle Packaging out of Alsip, Illinois for corrugate. It can't hurt to call.