Files grow at the end. You can't easily insert a line anywhere else. You might try playing with the buffer size to speed up the process, though:

$/ = \65535;  # You can try different numbers here.
print {$NEW} $header;
print {$NEW} $_ while <$OLD>;
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Update: it seems read is a bit faster on my system. YMMV.

print {$NEW} $_ while read $OLD, $_, 2 ** 30 - 1;
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