I wanted to keep the April 15 postmark, because I have great faith in the power of the Pittsburgh tax bureaucracy to get exquisitely fussy.
So I took it to the post office to find out whether this was a feasible thing to achieve with the mail. I was even considering hand-delivering the envelope to the tax office myself to preserve the postmark.
But it turned out to be much easier than that. All I needed to do was put the correct address on the envelope and hand it to the clerk; I didn't even need to put another stamp on it.
I am quite satisfied with the behavior of the post office in this.